Arithmetic Sequences

Transcription

1 . Arithmetic Sequeces COMMON CORE Learig Stadards HSF-IF.A. HSF-BF.A.1a HSF-BF.A. HSF-LE.A. Essetial Questio How ca you use a arithmetic sequece to describe a patter? A arithmetic sequece is a ordered list of umbers i which the differece betwee each pair of cosecutive terms, or umbers i the list, is the same. Describig a Patter Work with a parter. Use the figures to complete the table. Plot the poits give by your completed table. Describe the patter of the y-values. a. = 1 = = = = 5 y LOOKING FOR A PATTERN To be proficiet i math, you eed to look closely to discer patters ad structure. Number of stars, 1 5 Number of sides, y b. = 1 = = = = 5 y Number of circles, y c. = 1 = = = = 5 y 1 Number of rows, Number of dots, y Commuicate Your Aswer. How ca you use a arithmetic sequece to describe a patter? Give a example from real life.. I chemistry, water is called H O because each molecule of water has two hydroge atoms ad oe oxyge atom. Describe the patter show below. Use the patter to determie the umber of atoms i molecules. = 1 = = = = 5 Sectio. Arithmetic Sequeces 9

2 . Lesso What You Will Lear Core Vocabulary sequece, p. 1 term, p. 1 arithmetic sequece, p. 1 commo differece, p. 1 Previous poit-slope form fuctio otatio READING A ellipsis (...) is a series of dots that idicates a itetioal omissio of iformatio. I mathematics, the... otatio meas ad so forth. The ellipsis idicates that there are more terms i the sequece that are ot show. Write the terms of arithmetic sequeces. Graph arithmetic sequeces. Write arithmetic sequeces as fuctios. Writig the Terms of Arithmetic Sequeces A sequece is a ordered list of umbers. Each umber i a sequece is called a term. Each term has a specific positio i the sequece. Core Cocept 5, 1, 15,, 5,...,,... Extedig a Arithmetic Sequece Write the ext three terms of the arithmetic sequece. 7, 1, 1,,... 1st positio rd positio th positio Arithmetic Sequece I a arithmetic sequece, the differece betwee each pair of cosecutive terms is the same. This differece is called the commo differece. Each term is foud by addig the commo differece to the previous term. 5, 1, 15,,... Terms of a arithmetic sequece commo differece Use a table to orgaize the terms ad fid the patter. Positio 1 Term ( 7) +( 7) +( 7) Add 7 to a term to fid the ext term. Each term is 7 less tha the previous term. So, the commo differece is 7. Positio Term ( 7) +( 7) +( 7) The ext three terms are 5,, ad 9. Moitorig Progress Help i Eglish ad Spaish at BigIdeasMath.com Write the ext three terms of the arithmetic sequece. 1.,,,,.....,., 1, 1.,....,, 1, 1,... 1 Chapter Writig Liear Fuctios

3 Graphig Arithmetic Sequeces To graph a sequece, let a term s positio umber i the sequece be the x-value. The term is the correspodig y-value. Plot the ordered pairs (, ). Graphig a Arithmetic Sequece Graph the arithmetic sequece,,, 1,.... What do you otice? Make a table. The plot the ordered pairs (, ). Positio, Term, 1 1 The poits lie o a lie (, 1) (, ) (, ) (1, ) Idetifyig a Arithmetic Sequece from a Graph Does the graph represet a arithmetic sequece? Explai. Make a table to orgaize the ordered pairs. The determie whether there is a commo differece (1, 15) (, ) (, 9) (, ) Positio, 1 Term, ( ) +( ) +( ) Each term is less tha the previous term. So, the commo differece is (, 11) (, 7) (, ) (1, ) Cosecutive terms have a commo differece of. So, the graph represets the arithmetic sequece 15,, 9,,.... Moitorig Progress Graph the arithmetic sequece. What do you otice? Help i Eglish ad Spaish at BigIdeasMath.com.,, 9,,... 5.,,,,.... 1,.,.,., Does the graph show represet a arithmetic sequece? Explai. Sectio. Arithmetic Sequeces 11

4 Writig Arithmetic Sequeces as Fuctios Because cosecutive terms of a arithmetic sequece have a commo differece, the sequece has a costat rate of chage. So, the poits represeted by ay arithmetic sequece lie o a lie. You ca use the first term ad the commo differece to write a liear fuctio that describes a arithmetic sequece. Let a 1 = ad d =. ANOTHER WAY A arithmetic sequece is a liear fuctio whose domai is the set of positive itegers. You ca thik of d as the slope ad (1, a 1 ) as a poit o the graph of the fuctio. A equatio i poit-slope form for the fuctio is a 1 = d( 1). This equatio ca be rewritte as = a 1 + ( 1)d. Positio, Term, Writte usig a 1 ad d Numbers 1 first term, a 1 a 1 secod term, a a 1 + d + = 7 third term, a a 1 + d + () = 1 fourth term, a a 1 + d + () = 1 th term, a 1 + ( 1)d + ( 1)() Core Cocept Equatio for a Arithmetic Sequece Let be the th term of a arithmetic sequece with first term a 1 ad commo differece d. The th term is give by = a 1 + ( 1)d. Fidig the th Term of a Arithmetic Sequece STUDY TIP Notice that the equatio i Example is of the form y = mx + b, where y is replaced by ad x is replaced by. Write a equatio for the th term of the arithmetic sequece 1, 11,, 5,.... The fid a 5. The first term is 1, ad the commo differece is. = a 1 + ( 1)d Equatio for a arithmetic sequece = 1 + ( 1)( ) Substitute 1 for a 1 ad for d. = + 17 Simplify. Use the equatio to fid the 5th term. = + 17 Write the equatio. a 5 = (5) + 17 Substitute 5 for. = 1 Simplify. The 5th term of the arithmetic sequece is 1. Moitorig Progress Help i Eglish ad Spaish at BigIdeasMath.com Write a equatio for the th term of the arithmetic sequece. The fid a 5.., 5,, 7,... 9., 1,,, ,, 1,,... Chapter Writig Liear Fuctios

5 You ca rewrite the equatio for a arithmetic sequece with first term a 1 ad commo differece d i fuctio otatio by replacig with f(). f() = a 1 + ( 1)d The domai of the fuctio is the set of positive itegers. Writig Real-Life Fuctios Olie biddig for a purse icreases by $5 for each bid after the $ iitial bid. Bid umber 1 Bid amout $ $5 $7 $75 a. Write a fuctio that represets the arithmetic sequece. b. Graph the fuctio. c. The wiig bid is $15. How may bids were there? a. The first term is, ad the commo differece is 5. f() = a 1 + ( 1)d Fuctio for a arithmetic sequece f() = + ( 1)5 Substitute for a 1 ad 5 for d. f() = Simplify. REMEMBER The domai is the set of positive itegers. The fuctio f() = represets the arithmetic sequece. b. Make a table. The plot the ordered pairs (, ). Bid Bid umber, amout, Bid amout (dollars) c. Use the fuctio to fid the value of for which f() = 15. f() = Write the fuctio. Biddig o a Purse = Substitute 15 for f(). 1 = Solve for. (, 75) (, 7) (, 5) (1, ) 1 5 Bid umber There were 1 bids. Games Total cost 1 $7 $9 $11 $1 Moitorig Progress Help i Eglish ad Spaish at BigIdeasMath.com 11. A carival charges $ for each game after you pay a $5 etry fee. a. Write a fuctio that represets the arithmetic sequece. b. Graph the fuctio. c. How may games ca you play whe you take $9 to the carival? Sectio. Arithmetic Sequeces 1

6 . Exercises Dyamic Solutios available at BigIdeasMath.com Vocabulary ad Core Cocept Check 1. WRITING Describe the graph of a arithmetic sequece.. DIFFERENT WORDS, SAME QUESTION Cosider the arithmetic sequece represeted by the graph. Which is differet? Fid both aswers. Fid the slope of the liear fuctio. Fid the differece betwee the terms a ad a. Fid the differece betwee cosecutive terms of the arithmetic sequece. Fid the commo differece of the arithmetic sequece (5, 19) (, 1) (, 1) (, 1) Moitorig Progress ad Modelig with Mathematics I Exercises ad, write the ext three terms of the arithmetic sequece.. First term: Commo differece: 1. First term: 1 Commo differece: I Exercises 5 1, fid the commo differece of the arithmetic sequece. 5. 1, 1,,, , 15, 5, 1, ,,,,....,, 1,, , 5,.5,, , 7,, 11,... I Exercises 11 1, write the ext three terms of the arithmetic sequece. (See Example 1.) ,, 5,,.... 1,,,, , 1,, 1,... 1.,,,, , 1,.7,., ,, 1, 1,... I Exercises 17, graph the arithmetic sequece. (See Example.) 17.,,,, ,, 15,, ,, 5, 7,...., 19,, 5,... I Exercises, determie whether the graph represets a arithmetic sequece. Explai. (See Example.) (, ) (, ) (1, 1) (, 1) (1, 7) 5 (, 55) (, ) (, 5) (1, 5) (, 19) (, ) (, ) (, 1) (1, ) (, ) (, 1) I Exercises 7, determie whether the sequece is arithmetic. If so, fid the commo differece. 7. 1,, 9, 5,.... 5, 9, 1,,... 9.,,,,.... 7, 1, 75, 9, FINDING A PATTERN Write a sequece that represets the umber of smiley faces i each group. Is the sequece arithmetic? Explai. 1., 1, 9, 1 1,...., 5.5,.5,.75,... 1 Chapter Writig Liear Fuctios

7 . FINDING A PATTERN Write a sequece that represets the sum of the umbers i each roll. Is the sequece arithmetic? Explai. Roll 1 Roll Roll Roll REPEATED REASONING I Exercises ad, (a) draw the ext three figures i the sequece ad (b) describe the th figure i the sequece... I Exercises, write a equatio for the th term of the arithmetic sequece. The fid a 1. (See Example.). 5,,,,...., 9,, 15, , 1, 1 1,,.... 1, 11,, 1, ,, 1,,.... 7, 7, 5 7, 7, ERROR ANALYSIS Describe ad correct the error i fidig the commo differece of the arithmetic sequece., 1,, 1, The commo differece is 1.. ERROR ANALYSIS Describe ad correct the error i writig a equatio for the th term of the arithmetic sequece. 1,,,,... = a 1 + d = NUMBER SENSE The first term of a arithmetic sequece is. The commo differece of the sequece is 1.5 times the first term. Write the ext three terms of the sequece. The graph the sequece.. NUMBER SENSE The first row of a domioes display has 1 domioes. Each row after the first has two more domioes tha the row before it. Write the first five terms of the sequece that represets the umber of domioes i each row. The graph the sequece. 5. MODELING WITH MATHEMATICS The total umber of babies bor i a coutry each miute after midight Jauary 1st ca be estimated by the sequece show i the table. (See Example 5.) Miutes after midight Jauary 1st 1 Total babies bor a. Write a fuctio that represets the arithmetic sequece. b. Graph the fuctio. c. Estimate how may miutes after midight Jauary 1st it takes for 1 babies to be bor.. MODELING WITH MATHEMATICS The amout of moey a movie ears each week after its release ca be approximated by the sequece show i the graph. Earigs (millios of dollars) Movie Earigs (1, 5) 5 1 (, ) (, ) (, ) 1 5 Week a. Write a fuctio that represets the arithmetic sequece. b. I what week does the movie ear $1 millio? c. How much moey does the movie ear overall? Sectio. Arithmetic Sequeces 15

8 MATHEMATICAL CONNECTIONS I Exercises 7 ad, each small square represets 1 square ich. Determie whether the areas of the figures form a arithmetic sequece. If so, write a fuctio f that represets the arithmetic sequece ad fid f() REPEATED REASONING Firewood is stacked i a pile. The bottom row has logs, ad the top row has 1 logs. Each row has oe more log tha the row above it. How may logs are i the pile? 5. HOW DO YOU SEE IT? The bar graph shows the costs of advertisig i a magazie. Magazie Advertisemet. Cost (dollars) 7,, 5,,,, 1, 1 Size of advertisemet (pages) 9. REASONING Is the domai of a arithmetic sequece discrete or cotiuous? Is the rage of a arithmetic sequece discrete or cotiuous? 5. MAKING AN ARGUMENT Your fried says that the rage of a fuctio that represets a arithmetic sequece always cotais oly positive umbers or oly egative umbers. Your fried claims this is true because the domai is the set of positive itegers ad the output values either costatly icrease or costatly decrease. Is your fried correct? Explai. 51. OPEN-ENDED Write the first four terms of two differet arithmetic sequeces with a commo differece of. Write a equatio for the th term of each sequece. 5. THOUGHT PROVOKING Describe a arithmetic sequece that models the umbers of people i a real-life situatio. a. Does the graph represet a arithmetic sequece? Explai. b. Explai how you would estimate the cost of a six-page advertisemet i the magazie. 55. REASONING Write a fuctio f that represets the arithmetic sequece show i the mappig diagram PROBLEM SOLVING A trai stops at a statio every miutes startig at : a.m. You arrive at the statio at 7:9 a.m. How log must you wait for the trai? 57. ABSTRACT REASONING Let x be a costat. Determie whether each sequece is a arithmetic sequece. Explai. a. x +, x +, 5x +, 7x +,... b. x + 1, x + 1, 9x + 1, 7x + 1,... Maitaiig Mathematical Proficiecy Reviewig what you leared i previous grades ad lessos Solve the iequality. Graph the solutio. (Sectio.) 5. x < b. t 1 < y Graph the fuctio. Compare the graph to the graph of f(x) = x. Describe the domai ad rage. (Sectio.7). h(x) = x. v(x) = x 5. g(x) = x r(x) = x 1 Chapter Writig Liear Fuctios