SEQUENCES AND SERIES

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2 SEQUENCES AND SERIES U N I The umber of gifts set i the popular Christmas Carol days of Christmas form a sequece. A part of the sog goes this way O the th day of Christmas my true love gave to me drummers drummig 6 geese a layig pipers pipig 5 golde rigs 0 lords a leapig 4 callig birds 9 ladies dacig Frech hes 8 maids a milkig turtle doves, ad 7 swas a swimmig a partridge i a pear tree. How may gifts were give by the true love o the th day of Christmas?. The aswer is 78. How did I solve it? You will fid out as you lear the cocepts i oe of the lessos o arithmetic series. I this uit you will lear the differet sequeces ad series ad how they are applied i real life T VII

3 Lesso Sequeces A supermarket displays caed goods by stackig them, so that there are 0 rows of cas with cas i the top row. If each row, below the top row, had two more cas tha the rows just above it, how may cas could there be at the bottom row? Startig from the top row, the umber of cas i each row ca be listed as follows:, 5, 7, 9,,, 5, 7, 9, The stacks of cas are arraged i some order such that there are two cas more below each row. Ay such ordered arragemet of a set of umbers is called a sequece. The list of umbers, 5, 7, 9,,, 5, 7, 9, is called a sequece. Each of the umbers of a sequece is called a term of the sequece. The first term i the sequece is, the secod term is 5, while the third term is 7 ad the 0 th term is. Sequeces are classified as fiite ad ifiite. A fiite sequece cotais a fiite umber of terms. Examples:,,,, 5, 8,,, 4, 5,, 8, -,, - A ifiite sequece cotais a ifiite umber of terms. The ellipsis, is ofte used to show that a sequece is ifiite Examples:,, 5, 7,,,,, 4 8,,,, 5, 8, A sequece is a set of umbers writte i a specific order: a, a, a, a 4, a 5, a 6,, a. The umber a is called the st term, a is the d term, ad i geeral, a is the th term. You ca easily fid the ext term i a sequece by simply discoverig a patter as to how the terms are formed. You will fid that either a costat umber is added or subtracted or multiplied or divided to get the ext term or a certai series of operatios is performed to get the ext term.

4 Examples: Fid the ext term i each sequece.. 7,, 7,, Notice that 5 is added to get the ext terms or umbers i the sequece. 7 st term d term rd term th term What could be the ext term?.,,, 5 8 The fractios have as a commo umerator. The deomiators, 5, 8, form a sequece, by addig to the precedig umbers ( precedes 5, 5 precedes 8, 8 precedes ). What could be ext term after?. 5, 0, 0, 40, For this example, is multiplied by 5 to get 0, is multiplied by 0 to get 0 ad is also multiplied by 0 to get 40. So the ext term is 80, the result of multiplyig 40 by. 4. 4, -4, 4, -4, I this example, it is easy to guess that the ext term is 4 sice the terms i the sequece alterately are positive ad egative 4. Actually, the first, secod, ad third terms were multiplied by - to get the secod, third ad fourth terms, respectively. Let s Practice for Mastery : A. Write F if the sequece is fiite or I if the sequece is ifiite..,, 4, 5,.., 0. 7, 0,, 6, 9,, 5. 4, 9, 4, 9, 4., 4, 9, 6, 5,., 44 5., 4, 9 6, 4 5 4

5 Let s Do It : Why are Policeme Strog? Fid the ext umber i each sequece. Replace it with the letters o the left of each sequece. Write the letters that correspods to the sequece o the box below to decode the aswer to the puzzle. (Source: Math Joural) A, 5,,, N, 6, 8, 54, B, 4, 6, O 0, 9, 7, C 7,, 9, P,, 5, 7, 9,,, 5, D 9, 6,, R, 6, 9, E 4, 8, 0, 56, S 5, 7,,, F,, 4, 6, 0, 6, T,,, 4, 7,, 4, H,,, 4, 7,, U,,,,, 4, 6, 9,, I, 6,, 4, Y,,, 4,, 6, 4, 8, 5, 0, L 0,, 9,, 8, Let s Check Your Uderstadig : A. Write F if the sequece is fiite or I if the sequece is ifiite.., 6, 8, 54., 9, 7, 8,., 79,. -, 4, -8, 6, , 97, 94, 9,, - 5., 4, 8 6,, 64 B. Fid the ext term i the sequece of umbers i A. 5

6 Lesso The Terms of a Sequece Frequetly, a sequece has a defiite patter that ca be expressed by a rule or formula. I the simple sequece, 4, 6, 8, 0,. each term is paired with a atural umber by the rule a. Notice how the formula a gives all the terms of the sequece. Substitute,,, ad 4 for i a : Examples: a a a a a () a () 4 a () 6 a 4 (4) 8 Usig this formula, ca you fid a 0 or the 0 rd term of this sequece?. Fid the first four terms of the sequece whose geeral term is give by a. Solutio: To fid the first, secod, third ad fourth terms of this sequece, simply substitute,,, 4 for i the formula a -. If the geeral term is a, the the st term is a () d term is a () rd term is a () 5 4 th term is a 4 (4) 7. The first four terms of this sequece are the odd umbers,, 5, ad 7. The whole sequece ca be writte as,, 5,,. Write the first four terms of the sequece defied by a. + Solutio: Replacig with,,, ad 4, respectively the first four terms are: st term : a d term : a + + 6

7 rd term : a 4 th term: a The sequece defied by a + ca ow be writte as,,,,, Fid the st ( ) 5 terms of the sequece defied by a Solutio:. Agai by simple substitutio, st term : ( ) a d term: ( ) a rd term: ( ) a 4 th term: ( ) a th term: ( ) a The sequece defied by a ( ) ca be writte as ( ) -,, -,, -,, Notice that the presece of (-) i the sequece has the effect of makig successive terms alterately egative ad positive. Let s Practice for Mastery : A. Write the first four terms of the sequece whose th term is give by the formula.. a + 4. a ² +. a 5. a +. a 7

8 B. Fid the idicated term of the sequece whose th term is give by the formula.. a + 4 a. a a 8. a ( -) a 4. a (-) - ² a 5 5. a ( ) a 8 Let s Check Your Uderstadig : A. Write the first four terms of the sequece whose th term is give by the formula.. a 4. a -. a ² - 5. a ² -. a B. Fid the idicated term of the sequece whose th term is give by the formula.. a - 5 a 0 4. a ( ) a 5. a + a 5 5. a ( + )( + ) a 7. a + a Lesso Fidig the th Term of a Sequece I Lesso, some terms of a sequece were foud after beig give the geeral term. I this lesso, the reverse is doe. That is, give some terms of the sequece, we wat to fid a expressio for the th term. Examples:. Fid a formula for the th term of the sequece, 8, 8,, Solutio: Solvig a problem like this ivolves some guessig. Notice that the first four terms is twice a perfect square: () 8 (4) 8 (9) (6) 8

9 By writig each sequece with a expoet of, the formula for the th term is: a ()² a 8 ()² a 8 ()² a 4 (4)²... a ()² ² The formula of the th term of the sequece, 8, 8,, is a ².. Fid a expressio for the th term of the sequece, 8, 7 4, 64 5, Solutio: The first term ca be writte as. The deomiators, 8, 7, 64 are all perfect cubes, while the umerators are more tha the base of the cubes of the deomiators: + a a 8 a a Observig this patter, the expressio for the th term is a.. Fid the th term of the sequece -, 4, -8, 6, -, Solutio: These umbers are powers of ad they alterate i sig. The expressio for the th term is give by a (-) I,,,,,..., examie how the umerators ad deomiators chage as separate sequeces. Havig the kowledge of idetifyig the rule for the th term, + the expressio ow gives us a. 9

10 Not all sequeces ca be defied by a formula, like for the sequece of prime umbers. Let s Practice for Mastery : A. Fid a expressio for the th term for each of the followig sequeces:., 6, 9,, 4. 4, 9, 4, 9,.,, 7, 48, 5.,,, , 4, -8, 6, 4 6.,,, B. Fid the seveth term i each sequece.., 4, 6, 8, 4., 4, 9, 6,. 00, 50, 400, 5. 7, 7, 7, 7,., 0, 7, Let s Check Your Uderstadig : A. Fid a expressio for the th term for each of the followig sequeces:. 7, 0,, 6, 4, 6, 54, 8,. 4, 8,, 6, 0, 5.,,, , 4, 9, 6, 6.,,, B. Fid the seveth terms i each sequece.. 0,,,,, 4. -,, -, 4,.,, 6, 0, 5. 00, 00, 00, 0,.,, 4, 8, 6, Lesso 4 Arithmetic Sequeces Look at the followig sequeces.. 4, 7, 0,,., 8, 4, 48,. -, -6, -0, -4, 4. 00, 98, 96, 94, 5.,,,, Ca you give the ext two terms of each sequece above? How did you get the ext terms i each case? 0

11 If you get the ext two terms ad the umber added to the precedig terms to get the ext terms the you are correct:. ext two terms: 6, 9 the umber added:. ext two terms: 5, 58 the umber added: 5. ext two terms: -8, - the umber added:-4 4. ext two terms: 9, 90 the umber added: - 5. ext two terms:, the umber added:½ Notice that a fixed umber is added to the precedig term to get the ext term i the sequeces. These sequeces are called arithmetic sequeces. The fixed umber added is called the commo differece d. A sequece where each succeedig term is obtaied by addig a fixed umber is called a arithmetic sequece. The fixed umber is called the commo differece d. I order to idetify if a patter is a arithmetic sequece we must examie cosecutive terms. If all cosecutive terms have a commo differece you ca coclude that the sequece is arithmetic. Examples: Cosider the sequece of umbers. Fid the ext four terms of each.. 5, 5, 45, 65,., 9, 7, 5,. 0, 9, 8, 7, 4. -9, -4,, 6, I each sequece, how will you get the ext terms? First, fid the commo differece by subtractig the d term from the st, the rd from the d ad so o. I symbols, d a a-.. 5, 5, 45, 65, 5 5 0; ; The commo differece is 0. The fixed umber 0 is added to the precedig terms to get the succeedig terms.. 0, 9, 8, 7, 9 0 9; 8 9 9; The commo differece is 9.., 9, 7, 5,

12 9 8; 7 9 8; The commo differece is , -4,, 6, -4 (-9) ; (-4) + 4 5; 6 5 The commo differece is 5. Now, you are ready to fid the ext four terms beig asked for i the give sequece. See if you got these aswers.. 5, 5, 45, 65, 85, 05, 5, 45. 0, 9, 8, 7, 6, 45, 54, 6., 9, 7, 5,, 4, 49, , -4,, 6,, 6,, 6 How may correct aswers did you get? How did you get your aswers? Let s Practice for Mastery 4.: Determie whether the sequece is arithmetic or ot. If it is, fid the commo differece ad the ext three terms.., 5, 8,, 6., 5, 9,,., -4, 6, -8, 0, 7.,,,, , -0, -4, -8, 8. 5, 6, 7, 8, , 4, 44, 46, 9. 98, 95, 9, 89, 5..,.8,.4, ,,,, Let s Check Your Uderstadig 4.: Fid the commo differece ad the ext three terms of the give arithmetic sequece.., 0, 9, 8, 4.,, 5, 7,. 5.5, 7, 8.5, 0, 5. 4, 9, 5,.,,, 4, 6. 5, 4, 4, 5, Ca you give the 00 th term of the arithmetic sequece -8, -,, 7,? Kowig the commo differece you ca, but it would ot be easy listig the sequece of umbers from -8, our first term (a ) to the 00 th term or a 00.

13 You eed to use a equatio to defie the th term of the arithmetic sequece. Ay arithmetic sequece is defied by the equatio a a + ( )d, where, a is the th term, a is the st term ad d is the commo differece. Examples:. Fid the 5 th term of the arithmetic sequece for which the first term is 9 ad the commo differece is 7? a. Sice d 7 ad a 9, we ca easily fid the ext 5 terms of the arithmetic sequece as: 9, 6,, 0, 7 b. Give: a 9, 5, ad d 7 Fid: a 5 Substitute the give values i the formula: a a + ( )d a (5 )7 9 + (4) a 5 7 Therefore, 7 is the 5 th term or a 5 of the sequece.. Fid the 00 th term of the sequece i #. Give: a 9, 00, d 7 Fid: a 00 Substitute the give i the equatio: a a + ( )d a (00 )7 9 + (99) a Therefore, the 00 th term or a 00 is 70.. I the arithmetic sequece, 8, 5,,, which term equals 59? Give: a 59, t, d 7 Fid:

14 Substitute the give values i the formula: a a + ( )d 59 + ( ) Therefore, the 75 th term or a 75 of the arithmetic sequece is 59. Let s Practice for Mastery 4.: Fid the term idicated i each of the followig arithmetic sequeces.., 4, 6, 5 th term., 6, 9,, 5 th term. 99, 88, 77, 66, 8 th term 4. 99, 87, 75, 6, th term 5.,,,, 0 th term Let s Check Your Uderstadig 4.: Fid the term idicated i each of the followig arithmetic sequeces.. -8, -,, 7, rd term. 9, 84, 77, 70, 7 th term.,,, 4, 4 th term 4. 5, 4, 4, 5, 0 th term 5. 0, 4, -, -8, d term Lesso 5 Applicatios of Arithmetic Sequece i Real Life Without our kowig it, we are applyig the cocept of arithmetic sequece i real life. The examples below illustrate some of these applicatios.. Mrs. Lacso bought a house ad lot at the begiig of 995 for Php,500,000. If its value icreased by Php00,000 each year, how much was it worth at the ed of 005? I 995 the amout of the house ad lot bought by Mrs. See was Php,500,000. I the followig year, 996, Php00,000 was added to the origial amout, thus havig the ew value of Php,600, Let us tabulate to solve the problem. 4

15 Year # of years after Value of the House ad Lot purchase date 995 st,500, d,500, ,000,600, rd,600, ,000,700, th,700, ,000,800, th,800, ,000,900, th,900, ,000,000, th,000, ,000,00, th,00, ,000,00, th,00, ,000,00, th,00, ,000,400, th,400, ,000,500,000 The house ad lot is worth Php,500,000 at the ed of 005. Usig the formula a a + ( )d, we ca solve the problem as show below. a,500,000 + ( )00,000,500,000 +,000,000 a,500,000 Note that the resultig amout is the same as that i the table.. A restaurat has square tables which seat four people. Whe two tables are placed together, six people ca be seated as illustrated below. If 0 square tables are placed together to form oe log table, how may people ca be seated? If 00 square tables are placed together to form oe very log table, how may people ca be seated? You may use a table i order to see if there is a patter that relates the umber of tables to the umber of people that ca be seated. 5

16 Number of Tables placed together Diagram Number of Seats The umber of seats i the sequece begi with 4, 6, 8, 0,... To fid the umber of people that ca be seated at 0 tables that are placed together, we use the formula i fidig the th term of a arithmetic sequece, that is, a a + ( )d. Give: a 4, d, 0 Fid: a 0 Substitute the give values i the formula: a a + ( )d a (0 ) 4 + (9) a 0 4 Therefore, 4 people could sit at 0 tables. You ca fid the umber of people that ca be seated at 00 tables, usig the same formula. 6

17 Give: a 4, d, 00. Fid: a 00 Substitute the give i the formula: a a + ( )d a (00 ) 4 + (99) a 00 0 Therefore, 0 people could sit at 00 tables. Let s Practice for Mastery 5: Solve the followig problems.. The force of gravity causes a ball to fall 6. decimeters durig the first secod, 48. the ext secod, 80.5 the third, ad so o. How far will the body fall i 0 secods?. SEJ Compay offers you a job with choice of two salary icrease plas. With Pla A, you receive a aual salary of Php00, ad a aual icrease of Php With Pla B, you will receive a semiaual salary of Php50, ad a semiaual icrease of Php50. Which pla should you choose? How much moey will you receive over five years for each of the plas?. I a Math competitio, 0 questios were asked for each participat. For every wrog aswer, a competitor loses poit. If Lia eared 6 poits durig the first questio, ad the loses poit for the 9 remaiig questios. What was her score after the 0 th questio? Let s Check Your Uderstadig 5: Customers payig a electric bill were give tickets with a umber. The first customer got umber 04, the secod customer got 06, the third customer got 08, ad so o. The umbers o the ticket forms a arithmetic sequece. a. If the sequece cotiues, what umber will the 0 th customer get? b. What is a 0? c. Which customer got ticket umber 0400? d. If the last customer got ticket umber 664, how may customers etered the electric compay? 7

18 Lesso 6 Examples: Fidig the st Term ad the Commo Differece Give Two terms of Arithmetic Sequece. The rd term of a arithmetic sequece is 8 ad 7 th term is 0, fid the first term. We ca solve this i a simpler way. The sequece looks like the oes below. _, _, 8, _, _, _, 0 a a a 7 If we cosider 8 as our first term ad 0 as our th term, we ca use our formula: a a + ( )d Usig a 0, a 8 ad 5 (sice there are 5 terms from 8 to 0), the a a + ( )d (5 )d d 0 8 4d 4d d 4 d We ca ow use the value of d to solve for a i the origial problem. Substitutig a 7 0 ad d i the formula, we obtai a a + ( )d 0 a + (7 )() 0 a + 6() 0 a a a or a Let s Practice for Mastery 6: The 0 th ad th terms of a arithmetic sequece are ad 4, respectively. a. What is d? b. What is a? c. What is a 4? d. How may terms are egative? Let s Check Your Uderstadig 6: The 00 th ad 00 th terms of a arithmetic sequece are 8 ad 0, respectively. a. What is the first term? b. What is a 50? c. What is a 45? 8

19 Lesso 7 Solvig Problems Ivolvig Arithmetic Meas I a arithmetic sequece, the term(s) betwee ay two terms is (are) called arithmetic mea(s) betwee two terms. Examples: I the sequece, 6, 9,, the two arithmetic meas betwee ad are 6 ad 9.. Fid the arithmetic meas betwee ad 8. Give two arithmetic meas there are four terms i all. Assume that a ad a 4 8. Let us have the diagram of the sequece.,,, 8 a, a, a, a 4 a a + ( )d 8 + (4 )d 8 + d 8 d 6 d d Sice the first term, a, is give ad d, the it will be easy for us to fid the two arithmetic meas. Hece, a + a 4 a 4 + a 6 The umbers 4 ad 6 are the two arithmetic meas betwee ad 8. Fid the five arithmetic meas betwee 5 ad 47. Give five arithmetic meas there are seve terms i all. Assume that a 5 ad a Let us have the diagram of the sequece. 5,,,,,, 47 a, a, a, a 4, a 5, a 6, a 7 a a + ( )d (7 )d d d 9

20 4 6d d 7 Hece, a a a a a The umbers, 9, 6,, ad 40 are the five arithmetic meas betwee 5 ad 47.. Isert six arithmetic meas betwee ad 6. Also, prove that their sum is 6 times the arithmetic mea betwee ad 6.,,,,,,,6 a, a, a, a 4, a 5, a 6, a 7, a 8 Let a, a, a, a 4, a 5, a 6 be the six arithmetic meas betwee ad 6. The, by defiitio,, a,..., a 6, 6 are i arithmetic sequece. Let d be the commo differece. Here, 6 is the 8 th term. a a + ( )d a 8 + (8 )d 6 + (8 )d 6 +7d 6 7d 4 d 7 d Hece, the six arithmetic meas are 4, 6, 8, 0,, ad 4. Now the sum of these meas: Fid the arithmetic mea betwee ad 6. Let d be the commo differece. Here 6 is the rd term. a a + ( )d a + ( )d 6 + ( )d 6 +d 6 d 4 d d 7 Hece, the arithmetic mea betwee ad 6 is 9. The sum of the 6 arithmetic meas betwee ad 6, which is 54 is 6 times its arithmetic mea, 9. That is 54 6(9). 0

21 Let s Practice for Mastery 7: Solve what is asked:. Isert four arithmetic meas betwee - ad 4.. Isert five arithmetic meas betwee 4 ad 86.. Isert three arithmetic meas betwee -8 ad Isert four arithmetic meas betwee ad - 5. Isert oe arithmetic mea betwee 4 ad 68. Such a umber is called the arithmetic mea of the two umbers. Let s Check Your Uderstadig 7: Solve what is asked:. Fid the arithmetic mea of 7 ad -5.. Fid the four arithmetic meas betwee 7 ad -5.. Fid the arithmetic mea of 5 ad Isert 5 arithmetic meas betwee - ad 0. Show that their sum is 5 times the arithmetic mea betwee - ad Isert 0 arithmetic meas betwee -5 ad 7 ad prove that their sum is 0 times the arithmetic mea betwee -5 ad 7. Lesso 8 The Arithmetic Series Eight basketball teams are participatig i the summer sportsfest. If each of the team plays oce with each of the others, how may games will be played i all? Let us simplify the problem by represetig the 8 teams by letters A, B, C,, H. The game played by team A ad team B ca be represeted by a pair of letters, AB. Listig the possible games played, AB, AC, AD, AE, AF, AG, AH BC, BD, BE, BF, BG, BH CD, CE, CF, CG, CH DE, DF, DG, DH EF, EG, EH FE, FH EH 7 games 6 games 5 games 4 games games games game The total umber of games played by the 7 pairs of teams is

22 The idicated sum of the terms of this sequece is called a series. The idicated sum played by the 7 pairs of teams is illustrated as: S S + S + + S S S S Sice, we are iterested with the sum of the games played by the 7 pairs of teams the, S S 7 8. Notice, that the umber of games played by the teams form a arithmetic sequece. Thus, the idicated sum of the terms of this arithmetic sequece is called a arithmetic series ad sum S is called the value of the series. I the arithmetic sequece,, 5, 7, 9,,, the arithmetic series is \ S For a arithmetic series i which a is the first term, d is the commo differece, a is the last term, ad S is the value of the series, S ( a a + ) ad S [ a + ( ) d]. Examples:. Fid the sum of the first 5 odd umbers Let a, d, ad 5 i the formula S 5[() + (5 )] S 5 5 [ + (4)] 5 [ + 8] 5[0] 450 S 5 5 [ a + ( ) d]

23 . Fid the sum of the first 0 terms of the arithmetic sequece -5, -, -, Let a -5, d, ad 0 i the formula S 0[(5) + (0 )] S 0 5[-0 + (9)] 5[ ] 5 (8) S 0 40 [ a + ( ) d]. How may umbers betwee 0 ad 00 are exactly divisible by 7? Fid their sum. We kow that the first ad last umber betwee 0 ad 00 which is divisible by 7 is 0 ad 96. Hece, a 0, a 96 ad d 7. To fid, use a a + ( )d ( -) There are 7 umbers betwee 0 ad 00 that are divisible by 7. ( a ) To fid the sum use, S + a. Let s Practice for Mastery 8: ( a ) S + a 7 (4 + 96) S 7 7(0) 7(05) S 7,85 A. Fid the sum of the terms i the arithmetic sequece for the umber of terms idicated terms terms terms terms terms B. Fid the term asked by usig the give values.. a 45, a 7, d 9,

24 . a 79, a 7, d, S. a, d, 8, S d -5, a 7 -, 7, S 7 5. a 0 88, a -8, S 0 Let s Check Your Uderstadig 8: Solve the followig:. Fid the sum of the first 50 coutig umbers.. Fid the sum of the first 50 odd atural umbers.. Fid the sum of the first 4 terms of arithmetic sequece 5, 8,, 4, 7,..., 4. How may umbers betwee 5 ad 400 are multiples of? Fid their sum. 5. Fid the sum of all positive itegers betwee 9 ad 0 that are divisible by 4. Lesso 9 Applicatios i Real Life A school libraria purchases 0 assorted books durig the first moth of a cotract from a publisher, 5 i the secod moth, 0 i the third moth, 5 i the fourth moth, ad so o. The libraria wats to kow the total umber of books the school will have acquired after 0 moths. Note that the sequece is 0, 5, 0, 5, With d 5, a 0 ad 0, substitutig these i the formula S 0[(0) + (0 )5] we have S 0 5 [0 + (9)5] 5 [65] S [ a + ( ) d], Thus, after 0 moths the school will acquire,475 books. Let s Practice for Mastery 9: Solve the followig problems.. If a clock strikes the appropriate umber of times o each hour, how may times will it strike i oe day? I oe week?. A group of hikers has a trek of 6 days to reach Mt. Apo. They traveled 5 km o the st day, km o the d day, o the rd day, ad so o. How may kilometers did they travel to reach Mt. Apo?. Luis applied for scholarship ad was give battery of test. He made a score of 68 o his first test. The passig average score is 75. Would he make it after four test 4

25 if he did 6 poits better o each succeedig test? What was his score o the fourth test? What was his average score i the battery test? Let s Check Your Uderstadig 9: Solve the followig problems.. Jessie decided to do oe more pushup i his exercise each day tha he had doe the previous day. The first day he did 0 pushups. a. How may pushups did Jessie do o the 0 th day? b. How may pushups did Jessie do altogether i 0 days?. Athea vowed to study 4 hr more each day tha the previous day. The first day she studied 4 hr. a. How may hours did Athea study o the 0 th day? b. How may hours did Athea study altogether i 0 days? Lesso 0 The Geometric Sequece Makig oe fold o a sheet of a paper, we ca form two rectagles. Now let us fold the paper agai, ad cout the rectagles formed (cout oly the smallest rectagle as show below). Cotiue this process util you ca o loger fold the paper The umber of rectagles formed produces a geometric sequece,,, 4, 8, 6,... Notice each term after the first may be formed by multiplyig the previous term by. A geometric sequece is a set of terms i which each term after the first is obtaied by multiplyig the precedig term by the same fixed umber called the commo ratio which is commoly represeted by r. A sequece a is called geometric sequece if there is a o-zero umber r such that a r a, >. The umber r is called the commo ratio. 5

26 Here are some examples of a geometric sequece..,, 4, 8,. 9, -7, 8, -4,.., ,.05 4.,,,, Each is called a geometric sequece sice there is a commo multiplier or commo ratio, r, betwee the terms of the sequece after the st term. I (), the commo ratio, r, is ; i (), r -; i (), r 0.5: ad i (4), r. The commo ratio ca be foud by dividig ay term by its precedig term. The sequece, 6, 8, 54, 6, is a geometric sequece i which the first term, a, is ad the commo ratio is r a a a a 4 a a r 6 8 Example:. Fid r ad the ext three terms of the geometric sequece ,,,, 4 8 Solutio: To fid r, choose ay two cosecutive terms ad divide the secod by the first. Choosig the secod ad third terms of the sequece, a a r 4 5 To fid the ext three terms, multiply each successive term by a r a a 5 a5 - a a 6 a6 - a a 7 a7 - a The commo ratio is ad the ext three terms are,,

27 . Fid the first five terms of a geometric sequece whose first term is ad whose commo ratio is -. Solutio: Sice a ad r -, the a a r a - r a - -6 a r a - r a - (-6) 8 a 4 r a 4 - r a - (8) -54 a 5 r a 5 - r a 4 - (-54) 6 The first 5 terms of the sequece are, -6, 8, -54 ad 6. Let s Practice for Mastery 0. A. Tell whether the followig sequeces is geometric or ot. If geometric, fid r.. 4, 8, 6,, 4.,,, -, ,,,,, 5. 5, 5, 45, 5,., -, 9, -7, 8, B. Write the first five terms of the geometric sequece where,. a, r 4. a, r 4. a, r 5. a, r -. a 0, r, Let s Check Your Uderstadig 0.: A. Tell whether each sequece below is geometric or ot. If geometric, fid r.. 4 8,,,, 4., -4, 6, -64,., -, 7, -, 5. 0, 0, 0,.,,,, B. Write the first five terms of the geometric sequece where. a, r - 4. a -, r 0.5 7

28 . a, r - 5. a -, r -. a, r 0.5 Let s Do It 0.: What is the World s Fastest Isect? Aswer the followig problems o geometric sequece to fid out. Cross out the boxes that cotai a aswer. The remaiig boxes will spell out the ame of the world s fastest isect, which ca travel at a speed of aroud 60 kilometers per hour. Really amazig! (Source: Math Joural Vol. X, No. 4, 00 00). What is the commo ratio of the geometric sequece 8, 4, 79,...?. What is the missig term of the sequece 5, 5, 45,, 405,...?. What are the ext two terms of the sequece 7, 49, 4,...? 6 4. What is the commo ratio of the sequece,,, ? 6 5. Give the ext two terms of the sequece,,, Fid the missig term of the sequece,, ,,...? 6 B D U R A T 9 G 0 54 T R 40;6807 O 6 N F E 5 L 7 Y 40; ; Aswer: 8

29 Lesso The th term of a Geometric Sequece What if you are asked to fid, the 5 th term of a geometric sequece? Does it mea that you have to fid the 4 th term first to get the 5 th term? But, sice the 4 th term is ot give, you have to compute theth term to get the 4 th term. I other words, you have to get first all the terms precedig the 5 th term. Is there a shorter way of doig this? Actually, there is! There is a rule or formula for the th term of ay geometric sequece. Rule or Formula for the Geeral Term of a Geometric Sequece: If a is a geometric sequece with commo ratio, r, the a a r where is the umber of the term (term umber) ad a is the first term. A lot of problems ivolvig geometric sequece is solved usig this rule for the geeral term of a geometric sequece. You will see i the followig examples. Examples:. Fid the first five terms of a geometric sequece whose first term is ad whose commo ratio is -. Sice a ad r -, the proceed as follows a a a a 4 a a r a r a r (-) -6 a r a (-) (9) 8 4 a r 5 a r r r 4 r a (-) (-7) -54 a (-) 4 (8) 6 The first 5 terms of the sequece are, -6, 8, -54, ad 6. The rule for fidig the geeral term of a geometric sequece is a coveiet way for you to fid the th term of a geometric sequece. You do ot deped o the previous term to get the ext term.. Fid the th term of the geometric sequece whose first three terms are give below. 8 6 a. 4,, b. 5, -0, 0 9 9

30 Solutio: Sice the geeral term of a geometric sequece is a idetify the first term ad the commo ratio. a r, you have to The first term is: a. a 4 b. a 5 The commo ratio is ot give, so you have to fid the commo ratio by dividig a term by the precedig term. For this case, take a ad a so that a. r a 4 a b. r a a Replace a ad r ito the rule for the geeral term: a. a a r b. a 5 (-) - a 4 Notice that it is o loger possible to simplify further Let s Practice for Mastery.: A. Write the first five terms of the geometric sequece with the give st term ad commo ratio.. a 5 r. a r 4. a r a - r - 5. a.5 r 0.5 B. Fid the th term of the geometric sequece.., 8,,. -4,, -6,. 6, 4, 8, , 5,, , -,, 0

31 Let s Check Your Uderstadig.: A. Write the first five terms of the geometric sequece with the give first term ad commo ratio.. a r. a r 5. a 4 r a -5 r 5. a 0. r -0.5 B. Fid the th term of the geometric sequece.., 5, 5,. -, 6, -,. 8, 6, 9, ,,, ,,, , 0.05, 0.005, Let s Do It.: The World s Tallest Skyscraper The world s tallest skyscraper is 509 meters high. It is foud i Taipei ad is 0 stories high. It took over the title from the Petroas Twi Towers (45 m) i Kuala Lumpur which was cosidered the tallest buildig from 997 to 00. Fid the world s tallest skyscraper by aswerig the followig. Directios: Ecircle the letter that correspods to the correct aswer. The letters will spell out the ame of the skyscraper.. What is the third term of the geometric sequece, a (-) -? M. 8 T. 4 R. 8 S. -4. What is the secod term of the geometric sequece, a (-) -? A. H. 0 K. O.. The fourth term of the geometric sequece, a () - is. P. 6 S. 6 I. 54 U is the first term of the geometric sequece, a () - P. S. 6 T. 9 U.

32 5. I a 4 (4) -, a 4 is equal to. I. 4 M. N. 4 E I a (-4) -, 8 is the term. K. st L. d I. rd M. 4 th 7. I a a r -, if a 8 ad r, the what is a6? 56 8 I. B D. 4 6 E I a a r -, fid the eighth term of the geometric sequece whose first term is 64 ad whose ratio is -? P. 8 Q. 4 R. O Fid the commo ratio of the geometric sequece 8, 54, 6, 4. A. I. J. K. Aswer: Lesso The Geeral Term of a Geometric Sequece You are ow ready to apply the formula of a geometric sequece. As you go over the examples, the specific skills will be idetified. Examples: Fidig the Specific Term of a Geometric Sequece. Fid the sixth term of the geometric sequece, 6,, Solutio: Fid the commo ratio. r a 6 a Substitute i the formula for the th term of a geometric sequece with -6, a, r. a a r a 6 () 6- () 5 () 96

33 Notice that the problem simply asks for oly the 6 th term. This ca be solved by listig all the terms of the sequece. Thus, cotiuig the sequece will give the 6 th term. Thus, the 6 th term is 96. a, a, a, a 4, a 5, a 6, 6,, 4, 48, 96. Fid the 7th term of the geometric sequece whose first term is 4 ad whose commo ratio is -. Solutio: Sice a 4, r - ad the 7 th term is to be foud, use the formula a a r 7 a 7 a r a 7 4(-) 6 4(79) a 7 96 Fidig a Term, Give Two Other Terms of a Geometric Sequece. The third ad sixth terms of a geometric sequece are 5 ad -4, respectively. Fid the eighth term. Solutio: Notice that either the first term or the commo ratio is give i this problem. So the solutio for this oe is ot the covetioal way of solvig geometric sequeces. Let a 5 ad a 6-40, ad 6 i a a r a a r - ad a 6 a r 6- a a r a 6 a r 5 5 a r () -40 a r 5 () A system of equatios i two variables occurs. Recall that oe way of solvig a system of liear equatios i variables is by substitutio. To do this, solve for oe variable, i terms of the other. I this case, solve for a i terms of r, i equatio (). So that, 5 ar² 5 a r r r 5 r a Divide both sides by r.

34 5 The, replace a by r i equatio () 5 a r 5 r 5r r 5 5 r simplifyig r gives r divide both sides by r³ r - sice -8 is the third power of - Substitute - for r i ay of the two equatios to solve for the other missig variable, a. Usig equatio (), 5 a r² () 5 a (-)² 5 4 a 5 4a Divide both sides by a 4 Fially, solve the problem, that is, fid the 8 th term: a a r a 8 a r a 8 ( ) 7 a 8 ( 8) 5(-) -60 There is aother way of solvig the problem above without usig systems of equatios. It is give below. Solutio : Sice the geometric sequece gives the rd ad 6 th terms, it ca be writte as,, 5,,, -40,, a, a, a, a 4, a 5, a 6, a 7, a 8 Deletig the first two terms, aother geometric sequece is foud that begis as 5,,, -40,, a, a, a, a 4, a 5, a 6 4

35 Note that this secod geometric sequece has the same commo ratio as the origial geometric sequece. For this secod sequece, a 5 ad a Substitutig the formula a a r, the a 4 a r 4 - a 4 a r³ (r³) 40 5r Divide both sides by r³ - r Take the cube root of each side. Sice what is asked is the 8 th term i the origial sequece ad it has become the 6 th term i the secod sequece, solve for the 6 th term i the secod sequece. a 6 a r 6- a 6 a r 5 a 6 5(-) 5 a 6 5(-) a 6-60 Therefore, the 8th term i the origial sequece is -60. Notice that you do ot have to compute for the first term of the origial sequece sice what is asked is oly to fid the 8 th term. Fidig the Term Number (Positio) of a Term i a Fiite Geometric Sequece 4. I the geometric sequece whose first term is -5 ad whose commo ratio is -, which term is 0,40? Solutio: Let a -5, r - ad 0,40 as the th term i a a r 0,40-5(-) - 0,40 5( ) (-) - ( ) -048 Sice a a is the same as. ( ) a ( ) ( ) 048 ( ) Multiply each side by (-) (-) Sice 4096 is the th root of (-). Therefore, 0 40 is the th term. 5

36 Notice that the terms i a geometric sequece get quite large early i the sequece where the commo ratio is greater tha. The problem above ca the be solved by listig dow all the terms util the eeded term is obtaied. Solutio : Sice the st term is -5 ad the commo ratio is -, the by listig the terms of the problem above, -5, 0, -0, 40, -80, 60, -0, 640, -80, 560, -50, 040 a a 7 a It is easy to see that 0,40 is the th term. 5. Fid the commo ratio of a geometric sequece if the first term is ad the eighth term is Solutio: 87. Let a, ad a8 87 i a 8 a r a 8 r 87 7 r 7 87 r Multiply both sides by. 87 r 7 7 r Sice Therefore, the commo ratio is 7. Let s Practice for Mastery : A. Fid the idicated term for each geometric sequece.., 0, 50, a 0. -, -, -9, a 5.,,, a 4.,,,... 6 a 8 5. a 5, r 5 a 40 B. Solve as directed.. Fid the 8 th term of the geometric sequece 8, 4,,, 6

37 . Fid the 6 th term of the geometric sequece whose first two terms are 4 ad 6.. Fid the 0 th term of the geometric sequece whose 5 th term is 48 ad 8 th term is -84. Let s Check Your Uderstadig : A. Fid the idicated term for each geometric sequece.., 8,,. a ,,,... 4 a , -4,,... a , 5, -45,. a 7 5.,,,... a 9 B. Solve as directed.. Fid the st term i the geometric sequece where the 4 th term is 4 ad the 7 th term is.. I the geometric sequece 4, 64, 04,, which term is 6 44? Lesso Geometric Meas Whe the first ad the last terms of a geometric sequece are give, the terms betwee them are called the geometric meas. For example, the geometric meas of the geometric sequece, 6, 8, 54, 6 are 6, 8 ad 54. To solve for the geometric meas of a give geometric sequece, the formula for the th term of a geometric sequece is also used. Example:. Isert geometric meas betwee 4 ad 4. Solutio: Listig dow the geometric sequece will show that there are five terms, which meas that 5. So that a 5 4 ad a 4. 4,,,, 4 Substitutig i the formula: a a r a r r 7

38 4 4 4 r divide both sides by r 4 ± r sice 8 is the fourth power of ± Note that the commo ratio, r, takes two values, + ad -. So that there are two sets of geometric meas that ca aswer the questio. To get the desired geometric meas simply multiply the precedig terms by the commo ratio. For r, 4,, 6, 08, 4 For r -, 4, -, 6, -08, 4 Therefore, the geometric meas are ±, 6, ad ±08.. Isert four geometric meas betwee ad 96. Solutio: Listig dow the terms of the sequece gives,,,,, 96 Let 6, a 6 96 ad a i a a r a 6 a r 6 96 r r divide both sides by r 5 r sice is the fifth power of Sice r is the the aswer is show below., 6,, 4, 48, 96 The geometric meas are 6,, 4 ad 48.. Fid the geometric mea betwee ad 9. Solutio: Here oly a sigle term is asked. So that,, 9 Usig the formula for the th term of a geometric sequece, a a r 8

39 9 r 9 r 9 r divide both sides by 6 r ±4 r 6 is the secod power of ±4 Multiplyig the first term,, by the commo ratio, ±4, the computed geometric mea is either 48 or -48. Geometric Mea betwee Two Numbers If b, c ad d form a geometric sequece, the c is the geometric mea betwee b ad d. So that, c d c bd c ± b d b c Therefore, the geometric mea betwee two terms / umbers is the square root of the product of the two terms/ umbers. 4. Fid the geometric mea betwee the two umbers. a. 8 ad 7 b. -7 ad - Solutio: Substitutig i the formula for the geometric mea betwee two umbers c ± b d a. c ± 8 7 ± 576 ± 4 b. c ± ( 7)( ) ± 784 ± 8 Let s Practice for Mastery : A. Fid the geometric mea betwee the give two umbers.. 4 ad ad 6. 8 ad ad 44. ad 4 B. Do as directed. 5. Isert two geometric meas betwee 5 ad. 8 9

40 . Isert four geometric meas betwee 4 ad Isert four geometric meas betwee ad. 4 5 Let s Check Your Uderstadig : A. Fid the geometric mea betwee the give two umbers. 5. ad ad ad. ad B. Do as directed ad 5 7. Fid the geometric mea betwee 5 ad Isert three geometric meas betwee ad 8 7. Lesso 4 Geometric Series A frog leaps of the previous jump. If the frog s first leap is 7cm, fid the distace the frog has covered after 5 leaps. To fid the aswer to this oe, you have to go over with this lesso. For this lesso, oly fiite geometric series will be discussed. Ifiite geometric series are discussed i the ext lesso. The idicated sum of the terms of a geometric sequece is called a geometric series, it is deoted by S. I symbols, S a + a r + a r + a r + + a r - + a r - The sum of terms of a geometric sequece is give by: S a r ( r where, a first term of a geometric sequece ad r commo ratio, r ) 40

41 It is good to ote that r should ot be equal to sice if it is, the deomiator will be zero ad will ot make ay sese. But what if r, does it mea that a sum does ot exist? Of course, the sum exists. Fid out usig the cocepts you will lear i this lesso. Examples:. Fid the sum of the first six terms of the geometric sequece, 6,, 4, 6 Solutio: The commo ratio is. The sum of 6 terms is give by: a r S ( ) r 6 ( ) S 6 ( 64) ( 6) S Fid the sum of 0 terms of the sequece:,, 4, Solutio: I this case: a, r, ad 0 S S S S S 0 0 a r ( ) r S

42 Usig the formula for a geometric series may seem to be tedious but with practice ad a little patiece, it will tur out to be very easy.. Fid the sum of the idicated umber of terms i the give geometric sequece. a. a, r -, 9 c. a, r -, b. a 8, r -, 5 d. a 8, r -, 0 Solutio: a. b. c. d. S S S S a r ( r a r ( r a r ( r a r ( r ) ) ) ) 9 S 9 [ ( ) ] [ ( ) ] [ ] S 5 [ ( ) ] 8[ ( ) ] 8[ ] + S [ ( ) ] [ ] [ 0] S 0 [ ( ) ] [ ] 8[ 0] From Example, oe ca geeralize that if r -, the S a whe is odd or S 0 whe is eve. 4. Fid the sum of the geometric series: up to 5 terms Solutio: Usig the formula for the sum of a geometric series with a ad r 4. S a r ( r ) 5 [ (4) ] [ 04] 4 ( 0) 0 The sum is 0. Let s Practice for Mastery 4: A. Do as directed.. Fid the sum of the first 8 terms of the geometric sequece:, 4, 8, 6,. What is the sum of the first terms of the geometric sequece.,,,,...? 4 8. What is the sum of the first 8 terms of,,,,...? 4 8 4

43 4. Fid the sum of the first 4 terms of the geometric series: Fid the sum of the first 7 terms of the geometric sequece:, 9, 7, 8, B. Fill i the table with the values that will make each a geometric sequece. No. a r a S Lesso 5 Sum of a Ifiite Geometric Sequece Sum of a ifiite geometric sequece! Is there such a thig? Well, there is! You are actually goig to lear it i this lesso. As a itroductio, let us start from what we kow. Earlier, we leared that the formula for the sum of a fiite geometric series is S a r ( r ) For example, i the sequece is 6,, 4, Sice r, as icreases, the value of r also icreases ad so does the sum, S. Each ew term adds a larger ad larger amout to the sum ad so there is o limit to the value of S ad S α does ot exist. A similar situatio occurs if r, so that geerally the sum to ifiity of a geometric sequece is S α a r The sum of the terms of a ifiite geometric sequece with first term a ad commo ratio r, where r <, is () a Sα or () S α a r r 4

44 Examples:. Fid the sum to ifiity of the geometric sequece with a 5 ad r -. Solutio: Substitutig the give values to the formula above, the sum is a 5 S α r ( ) S Fid the sum to ifiity of the geometric sequece 0, 5,,, Solutio: The commo ratio is 4 ad a 0. Substitutig i the formula S α S α a r For the ext example, the secod formula will be used.. Fid the sum to ifiity of the geometric sequece,,, Solutio: Substitutig i the secod formula, where a ad r, 4 the S α a r 44

45 45 Now, look at the solutio usig the first form of the formula: α S r a A thorough kowledge of all skills related to fractios helps i the uderstadig of how the solutio is doe. Let s Practice for Mastery 5 Fid the sum to ifiity of the geometric sequeces give below.., 4, 4, 6. a 000, r , 9, 0.09, 7. 6, 4, 6,. 8, 6,, 8. a 8, r a, r - 9. a 0, r , 4,, 0. a 8 9, r Lesso 6 Applicatios of Geometric Sequeces ad Series A lot of problems ca be solved by usig the formulas for the geeral term of a geometric sequece ad geometric series, fiite or ifiite. Of these applicatios, that of the ifiite geometric series is most iterestig as see i the followig examples. Examples: A. Chagig Repeatig Decimals to Fractios:. Show that the repeatig, o- termiatig decimal 0.77 is equal to.

46 Solutio: The decimal ca be writte as Writig the decimal as a fractio gives The series of umbers really is a ifiite geometric series, sice there is a 7 commo ratio, r, with a. So solvig for the sum, gives S α a r Hece, B. Chai Letter Problem Solutio:. Lida wrote a letter ad seds it to three frieds. Each of the three frieds writes the same letter ad seds it to other frieds ad the sequece is repeated. Assumig that o oe breaks the chai, how may letters will have bee set from the first through the sixth mailigs? The diagram will help i uderstadig the problem. st d O the first mail, letters are set, o the secod mailig there are () 9 letters set, o the third mailig there are 9() 7 letters set, ad so o. Observe that the sequece formed is, 9, 7, 46

47 The problem asked for the total umber of letters mailed. So the formula for the sum of terms of a geometric sequece is used. S a r ( r 6 ( ) S 6 ) ( 79) ( 78) 84 S 6 09 There are 09 letters mailed i all. C. Growth of Bacteria:. A certai culture of bacteria iitially cotais 000 bacteria ad doubles every hour. How may bacteria are i the culture at the ed of 0 hours? Solutio: Sice the umber of bacteria doubles every hour ad there are iitially 000, the at the ed of the first hour there will be 000. At the ed of the secod hour, there will be ad so o. A table of values is show below. t hours 4 5 o. of bacteria The secod row of the table shows a geometric sequece where a 000 ad r. Usig the formula for the th term of a geometric sequece, the, a a r 0 000() 000() 9 000(5) a There are bacteria at the ed of 0 hours. Notice that we did ot start the sequece with 000 sice it is the iitial umber of bacteria i the culture at t 0. The doublig starts at the ed of the first hour. 47

48 Let s Practice for Mastery 6: A. Write each of the followig repeatig decimals as a equivalet fractio: B. Solve the followig.. O the first swig, the legth of the arc through which a pedulum swigs is 0dm. The legth of each successive swig is 5 4 of the precedig swig. What is the total distace traveled by the pedulum has traveled durig the four swigs?. What distace will a golf ball travel if it is dropped from a height of 7 dm, ad if, after each fall, it rebouds of the distace it fell?. A culture of bacteria doubles every hours. If there are 500 bacteria at the begiig, how may bacteria will there be after 4 hours? 4. A particular substace decays i such a way that it loses half its weight each day. If iitially there are 56 grams of the substace, how much is left after 0 days? C. The followig is the Tower of Haoi Puzzle. Read it ad try to do what you are asked. The aswer the questios that follow. The Tower of Haoi is a puzzle that has the followig form: Three pegs are placed i a board. A umber of disk graded i size are staked i oe of the pegs with the largest disk at the bottom ad the succeedig smaller disk placed o top. The disks are moved accordig to the followig rules: a. Oly oe disk at a time may be moved. b. A larger disk caot be placed over a smaller disk. The object of the puzzle is to trasfer all the disks from oe peg to oe of the other two pegs. If iitially there is oly oe disk, the there will be oly oe move. With three disks, the oly oe move would be required. You ca try this puzzle usig cois of differet sizes. 48

49 The chart below shows the miimum umber of moves required for a iitial umber of disks. The differece betwee the umbers of moves for each succeedig disk is also give. No. of disks No. of moves x y p q r Questios:. What kid of sequece is the last list of umbers i the chart?. Fid the values of p, q ad r.. The fid x ad y. 4. What is the geeral term for the sequece of umbers i the secod row? 49

50 LET S SUMMARIZE A sequece is a set of umbers writte i a specific order: a, a, a, a 4, a 5, a 6,, a. where, a is called the st term, a is the d term, ad i geeral, a is the th term. Arithmetic sequece is a sequece where each succeedig term is obtaied by addig a fixed umber. Commo differece d is the fixed umber betwee ay two succeedig terms. The terms of a arithmetic sequece are defied by usig the formula a a + ( )d. Arithmetic series is a idicated sum of the first terms of a arithmetic sequece. The sum of terms is deoted by S. For a arithmetic series i which a is the first term, d is the commo differece, a is the last term, ad S is the sum of the series, ( a ) S + a [ a + ( ) d] ad S. A sequece a is called geometric sequece if there is a o-zero umber r such that a r a -, >, such that the umber r is called the commo ratio. If a is a geometric sequece with commo ratio, r, the a umber of the term (term umber) ad a is the st term. a r,where is the If b, c ad d form a geometric sequece the c is the geometric mea betwee b ad d. Thus, c d c bd c ± b d b c The idicated sum of a geometric sequece is called a geometric series. The sum of terms of a geometric sequece or the sum of a geometric series is give by the formula a r S ( ) r where a the first term, the umber of terms ad r the commo ratio. The sum of a ifiite geometric sequece or of a ifiite geometric series is give by the formula a Sα a r r where a the first term ad r the commo ratio such that r <. 50

51 Uit Test Aswer the followig:. Write the first five terms of the sequece a 5.. Fid a 8 i a What is the geeral term for 0, -4, -8, -? 4. For the sequece deoted by a Fid the first three terms of the sequece a Give the arithmetic sequece whose 7 th term is ad whose th term is Fid the three arithmetic meas betwee 9 ad. 8. Fid the 5 th term of the arithmetic sequece, 5, 8,,. 9. Fid the arithmetic mea of ad. 0. Fid the sum of the first 0 terms i the arithmetic sequece 0,,,,. What is the sum of the umbers from to 00?. How may umbers betwee 00 ad 400 are divisible by 5?. Which of the followig is a geometric sequece? 4. Fid the ext four terms i 0,,,, Fid the commo ratio of the geometric sequece, 6,, 4, 6. What is the geeral term of the sequece i umber 5? 7. Fid the 8 th term of the sequece, 6, 8, 8. Fid the ext six terms of the geometric sequece whose a 5 ad r Isert two geometric meas betwee 8 ad Fid the sum of the first terms of the geometric sequece, 6,.. Fid a equivalet fractio of the repeatig decimal 0.88,. Rosel started a chai of ispiratioal text messages to two of her frieds. Each of the two frieds seds the message to two other frieds ad the sequece cotiues. About how may text messages would have bee set after the sixth sedig?. A particular substace decays i such a way that it loses half of its weight each day. If iitially there are 56 grams of the substace, how much is left after 0 days? 4. Marc plaed to o a holiday i December at oe of the beach resort i Boracay. He started to save Php00 durig the moth of March ad each moth thereafter, doubles the amout he saved the moth before. How much would have bee saved by December? 5. Eda eeds Php to buy a bicycle. She has already saved Php50. If she saves Php00.00 a week from her job, i how may weeks must she work to have eough moey to buy the bicycle? 5

52 ANSWER KEY Lesso : Let s Practice for Mastery. F. F. I 4. F 5. F Let s Do It Because he ca hold up traffic Let s Check Your Uderstadig A. B.. F. 6. I. 87. I F F 5. 8 Lesso : Let s Practice for Mastery A. B..,, 4, , 5, 8,. -4., 4, 8, , 5, 0, ,,, Let s Check Your Uderstadig A. B.. 0,,,. 5. 0,, 8, 5.., 9, 7, , -, -5, , 7, 6,

53 Lesso : Let s Practice for Mastery A. B.. a. 4. a a (-) a a a ( ) + Let s Check Your Uderstadig A. B.. a a a a a a + Lesso 4: Let s Practice for Mastery 4.. Arithmetic; d ; 4, 7, 0 6. Arithmetic; d 4; 7,, 5. No 7. No. Arithmetic; d -4; -, -6, No 4. Arithmetic; d ; 48, 50, 5 9. Arithmetic; d -; 86, 8, Arithmetic; d 0.6;,.6, Arithmetic; d ; 7, 8, Let s Check Your Uderstadig 5. d 9; 7, 46, 55. d.5;.5,, 4.5. d 5, 6, 7 4. d ; 9,, 5. d -4;, 7, 6. d 9; 6, 70, 79 Let s Practice for Mastery 4.. a 5 0. a a