3n

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1 Prctice Set 6 Sequeces d Series Clcultor Required Objectives Alyze ptters i sequeces to determie subsequet terms. Fid the first four terms of sequece give equtio for. Fid expressio for give sequece. Expd series give i sigm ottio. Rewrite give series usig sigm ottio. Notes Sequece list of umbers Fctoril! o!... 0!!!! 6 4! 4 4 o 5! ! Series the sum of the terms i sequece o o For problems - you eed to kow the below iformtio. Note: The iformtio below will NOT be give o the ssessmet. Expoetil Fuctios x f x b c 0 grph 'orml' b expoetil growth (icresig) 0 grph reflected over x-xis 0 b expoetil decy (decresig) f x c horizotl symptote x xis if c 0 Qudrtic Fuctios f x x h k f x x bx c 0 cocve (opes) up 0 cocve (opes) up 0 cocve (opes) dow 0 cocve (opes) dow h, k vertex b b, f vertex c verticl itercept Sequeces d Series Ivestigtios

5 5 D. P.6 t 4.5 500. (ACT/SAT) Multiple choice clcultor required - A chemist is mesurig the lifetimes of umber of toms of the elemet medelevium.

2 . (ACT/SAT) Multiple choice - clcultor required The popultio, P, of formerly edgered species of wolf ws recorded every three moths for 5 yers. The sctterplot show below i the tp-ple shows the popultio t moths fter Jury, 990. Which of the followig equtios best models the reltioship betwee P d t? A. t P B. t P C. P.6 t D. P.6 t (ACT/SAT) Multiple choice clcultor required - A chemist is mesurig the lifetimes of umber of toms of the elemet medelevium. The plot show below i the LN-ple shows the umber of toms, N, which hd lifetime of L hours rouded to the erest hour. Which of the followig equtios best models the reltioship betwee N d L? A. L N 4. B. L N C. N 5 0.6L D. N 5.6L. (ACT/SAT) Wht is the sum of ll the possible vlues for z tht stisfy 5 z 4? z z 4. (ACT/SAT) Wht is the sum of the solutios to 4x 0 x? Sequeces d Series Ivestigtios

3 5. (ACT/SAT) The expressio 4 6x 8 d d re positive itegers. Wht is the vlue of b c? x b x b cx d, where, b, c, c be fctored ito the form 6. (ACT/SAT) Multiple choice Which of the followig is equivlet to b b c d d c for c d? A. 0 B. b c d C. b c d D. b c d For problems 7-8, solve for x. 7. (55) x e 4 8. (55) lx 0 For problems 9-, fid f ' x. x x 9. (56) f x e 0. (56) f x e. (56) f x e l x. (56) f x l x For problems -6, fid the ext two terms of ech sequece by lyzig the ptter of differeces.., 5, 9, 5,, 4. 48, 6, 6, 8,, 5.,, 7, 5,, 6.,, 6,, 9,, For problems 7-8, fid the first four terms of ech sequece Sequeces d Series Ivestigtios

4 0...! !! For problems 9-40, give ech sequece below, fid expressio for. 9. 0, 8, 6, 4, ,,,, ,,,, , 5, 45, 5, ,,,, ,, 4, 5, ,,,, ,,,, ,,,, ,,,, ,,,, 40.,,, 4, 5 For problems 4-49, write ech series i expded form Sequeces d Series Ivestigtios

5 For problems 50-6, rewrite ech series usig sigm ottio Sequeces d Series Ivestigtios 5

Prctice Set 6 Arithmetic d Geometric Sequeces Clcultor Required Objectives Fid the commo differece of rithmetic sequece or the commo rtio of geometric sequece.

6 Prctice Set 6 Arithmetic d Geometric Sequeces Clcultor Required Objectives Fid the commo differece of rithmetic sequece or the commo rtio of geometric sequece. Express rithmetic or geometric sequece s i terms of. Fid prticulr term of rithmetic or geometric sequece. Notes Arithmetic Sequece dd to get the ext term (commo differece d) Geometric Sequece multiply to get the ext term (commo rtio r) d Geometric - r th term: Arithmetic - For problems - you eed to kow the below iformtio. Note: The iformtio below will NOT be give o the ssessmet. Expoetil Fuctios x f x b c 0 grph 'orml' b expoetil growth (icresig) 0 grph reflected over x-xis 0 b expoetil decy (decresig) f x c horizotl symptote x xis if c 0 Qudrtic Fuctios f x x h k f x x bx c 0 cocve (opes) up 0 cocve (opes) up 0 cocve (opes) dow 0 cocve (opes) dow h, k vertex b b, f vertex c verticl itercept. (ACT/SAT) Multiple choice - clcultor required The sctterplot below shows the price, c, per troy ouce of gold i Uited Sttes (US) dollrs from 996 to 0, where t represets yers sice 996. Which of the followig equtios best models the dt? A. c 50.5 t B. c t C. c 0.4t D. c 0.4 t Sequeces d Series Ivestigtios

. (ACT/SAT) Multiple choice clcultor required - The sctterplot below shows the umber of times, s, i thousds, video hs bee shred o give dy, t, fter the video ws iitilly shred.

7 . (ACT/SAT) Multiple choice clcultor required - The sctterplot below shows the umber of times, s, i thousds, video hs bee shred o give dy, t, fter the video ws iitilly shred. If the expoetil fuctio tht best models the dt is show bove, which of the followig best describes the percet by which the umber of shres icresed ech dy? A. 5% B. 0% C. 8% D. 40%. (ACT/SAT) Wht is the sum of ll the possible vlues for tht stisfy 4 4 4? 4. (ACT/SAT) Wht is the sum of the solutios to x 0 x? 5. (ACT/SAT) Multiple choice - Which of the followig is equivlet to p 7 p 7? 7 7 A. p p 7 B. p p 7 C D (ACT/SAT) Multiple choice The expressio x 5x v c be writte s re costts. Wht is the vlue of? A. 5 B. 4 C. 4 D. 5 x 5x 5 where d v For problems 7-8, solve for x. 7. (55) x 6 e 8. (55) lx l9x Sequeces d Series Ivestigtios 7

8 For problems 9-, fid f ' x. 4x l x 9. (56) f x e 4 0. (56) f x e x. (56) f x lsec x. (56) f x l0 For problems -8: Fid the first four terms of the give sequece. Stte whether the sequece is rithmetic (stte the commo differece), geometric (stte the commo rtio), or either si 8. For problems 9-4, fid formul for. 9., 4, 7, 0, 0. 8, 6, 4,,. 8, 4,,,. 0., 0.9,.7, 8.,. 8, 0,, 44, 4. 84, 9, 96, 48 For problems 5-0: Stte whether the sequece is rithmetic (stte the commo differece), geometric (stte the commo rtio), or either. Fid formul for. 5. 7,, 5, 9, 6. 8,, 8, 7, 7., 4, 9, 6, 8 Sequeces d Series Ivestigtios

9 / 5/ 8/ / 8., 0, 00, 000, 9. b, b, 4, 5 + b, 0.,,,, For problems -8, fid the idicted term of ech sequece.. Arithmetic - 5,, 0?. Arithmetic - 76, 70, 0?. Arithmetic - 8, 5 4, 50? 4. Arithmetic - 8 5, 0 6,? / 5. Geometric -,,? 6. Geometric -,,? Geometric - 64, 5 8, 9? 8. Geometric - 8, 4 4, 7? 9. How my terms re i the rithmetic sequece, 8, 4,, 6? 40. How my terms re i the rithmetic sequece, 78, 70,? Sequeces d Series Ivestigtios 9

Prctice Set 6 Fiite Arithmetic d Geometric Series Clcultor Required Objectives Fid the sum of fiite rithmetic series. Fid the sum of fiite geometric series.

10 Prctice Set 6 Fiite Arithmetic d Geometric Series Clcultor Required Objectives Fid the sum of fiite rithmetic series. Fid the sum of fiite geometric series. Notes Arithmetic - S Geometric - r S r For problems - you eed to kow the below iformtio. Note: The iformtio below will NOT be give o the ssessmet. Expoetil Fuctios x f x b c 0 grph 'orml' b expoetil growth (icresig) 0 grph reflected over x-xis 0 b expoetil decy (decresig) f x c horizotl symptote x xis if c 0 Qudrtic Fuctios f x x h k f x x bx c 0 cocve (opes) up 0 cocve (opes) up 0 cocve (opes) dow 0 cocve (opes) dow h, k vertex b b, f vertex c verticl itercept. (ACT/SAT) Multiple choice - clcultor required A cliic recorded ge d boe mierl desity for severl of its ptiets i the sctterplot below. Give tht x represets the ge of the ptiet, i yers, d y represets the boe mierl desity, i grms g per squre cetimeter, which of the followig equtios best models the dt? cm A. y x. B. y x. C. y x. D. y x. 0 Sequeces d Series Ivestigtios

. (ACT/SAT) Multiple choice clcultor required - The plot below i the sp-ple shows the power, P, i kilowtts, geerted from wter flowig pst turbie with speed of s miles per hour.

11 . (ACT/SAT) Multiple choice clcultor required - The plot below i the sp-ple shows the power, P, i kilowtts, geerted from wter flowig pst turbie with speed of s miles per hour. Which of the followig equtios best models the reltioship betwee P d s? A. P s B. P s C. P 0.s 9 8 D. P 0. s 8 9. (ACT/SAT) Wht is the sum of ll the possible vlues for tht stisfy 4 4 4? 4. (ACT/SAT) Wht is the sum of the solutios to x 7 x? 5. (ACT/SAT) Give x x x x 5 for ll x, wht is the vlue of? 6. (ACT/SAT) Multiple choice Which of the followig is equivlet to x if for ll m? x 4x 4 m 4 4 m 4 m is true A. m B. m C. m D. m Sequeces d Series Ivestigtios

12 For problems 7-8, solve for x. 7. (55) l lx 8. (55) lx l4 lx For problems 9-, fid f ' x. 7x 9. (56) f x 4e 0. (56) f x x x e. (56) f x l x. (56) x 4 f x l x For problems -, fid the sum of ech rithmetic series.. S 0 :, S 40 :, S 5 : 8, S :, Sequeces d Series Ivestigtios

13 For problems -0, fid the sum of ech geometric series.. S 0 :, r 4. S 0 :, r 5. S : S 0 : Roud to the erest whole umber S :, 0. S 5 :, Sequeces d Series Ivestigtios

Prctice Set 64 Ifiite Series Clcultor Required Objectives Fid the sum of ifiite series, if it exists. Apply the ifiite geometric series to modelig scerios.

14 Prctice Set 64 Ifiite Series Clcultor Required Objectives Fid the sum of ifiite series, if it exists. Apply the ifiite geometric series to modelig scerios. Notes S if r r For problems - you eed to kow the below iformtio. Note: The iformtio below will NOT be give o the ssessmet. Expoetil Fuctios x f x b c 0 grph 'orml' b expoetil growth (icresig) 0 grph reflected over x-xis 0 b expoetil decy (decresig) f x c horizotl symptote x xis if c 0 Qudrtic Fuctios f x x h k f x x bx c 0 cocve (opes) up 0 cocve (opes) up 0 cocve (opes) dow 0 cocve (opes) dow h, k vertex b b, f vertex c verticl itercept. (ACT/SAT) Multiple choice - The plot below i the tv-ple shows the vlue, V, i thousds of dollrs, for portfolio of stocks d bods t yers fter purchse. Which of the followig equtios best models the reltioship betwee V d t? A. t V B. t V C. t V. 0 D. t V 0. 4 Sequeces d Series Ivestigtios

. (ACT/SAT) Multiple choice The plot below i the vp-ple shows the pressure, p, i kilopscls, of the iterl wter withi pipe mesured s the wter moves with prticulr velocities, v, i meters per secod

15 . (ACT/SAT) Multiple choice The plot below i the vp-ple shows the pressure, p, i kilopscls, of the iterl wter withi pipe mesured s the wter moves with prticulr velocities, v, i meters per secod flowig to the right. Which of the followig equtios best models the reltioship betwee p d v? A. p 00.7v B. p 00.7v C. p 0 0.5v D. p 0 0.5v. (ACT/SAT) Wht is the sum of ll the possible vlues of k tht stisfy 5 k? k k 4. (ACT/SAT) Wht is the sum of ll the solutios to 6x 50 4x 0? 5. (ACT/SAT) The equtio 4 z z is true for ll vlues of z. Wht is the vlue of t? t 0 6. (ACT/SAT) Wht is the vlue of k i 4 y y 56 k y y 56 k y 6 00? Sequeces d Series Ivestigtios 5

16 For problems 7-8, solve for x. 7. (55) l4x 5 lx 8. (55) 0 l x 0 For problems 9-, fid f ' x. x x 9. (56) f x 6 si e 0. (56) f x te. (56) f x l x. (56) f x l x x For problems -4, fid the sum if the series hs oe. If it does ot, stte tht the series diverges Sequeces d Series Ivestigtios

17 Ech yer certi type of tree grows verticlly 4 s much s it did the yer before. If the tree grows.5 5 feet durig the first yer, how my feet tll will it ultimtely grow? 6. A side of squre is cm. The midpoits of its sides re joied to form iscribed squre, d this process is cotiued. Fid the sum of the res of the squres. Problems 7-8 use the followig scerio: A side of equilterl trigle is 0 cm. The midpoits of its sides re joied to form iscribed equilterl trigle d the process is cotiued without ed. 7. Fid the sum of the perimeters of the trigles. 8. Fid the sum of the res of the trigles. 9. A bll dropped from height of 40 cm rebouds o ech bouce to height which is 5 of its height o the previous bouce. How fr will it trvel before comig to rest? 0. A bll is throw up height of 0 cm before fllig bck dow to the groud. The bll rebouds to height which is 8 of its height o the previous bouce. How fr will it trvel before comig to rest? Sequeces d Series Ivestigtios 7

Which of the followig best models the reltioship betwee N d t? A. N 5 0.56 t B. N 5.5 t C. N.68.8 t D. N.68 9. t. (ACT/SAT) Multiple choice The sctter plot below depicts the chge i prime ledig rtes (iterest rtes), y, of Uited Sttes (US mjor bks from 004 to 009.

18 Prctice Set 65 Assessmet 5 Review 60 Poits Clcultor Required. (ACT/SAT) Multiple choice - The sctterplot below shows the umber of smrtphoes sold, N, i millios, t certi compy from 008 to 0, where t represets yers sice 008. Which of the followig best models the reltioship betwee N d t? A. N t B. N 5.5 t C. N.68.8 t D. N t. (ACT/SAT) Multiple choice The sctter plot below depicts the chge i prime ledig rtes (iterest rtes), y, of Uited Sttes (US mjor bks from 004 to 009. If x is the umber of yers sice 004, which of the followig fuctios best describes the reltioship show? A. y x B. y x C. y 0.65x.74x.7 D. y 0.65x.74x.7. (ACT/SAT) Multiple choice - clcultor required Wht is the sum of ll the vlues of x tht stisfy x 4? x x A. B. C. D. 5 8 Sequeces d Series Ivestigtios

19 4. (ACT/SAT) Wht is the sum of ll the solutios to w 8 w 4 w? 5. (ACT/SAT) Multiple choice - Which of the followig is equivlet to b c b c? A. b c bc B. b c bc C. b c D. b c m 6. (ACT/SAT) Multiple choice Which of the followig is equivlet to m? A. m B. m C. m D. m m Sectio 55 The Nturl Logrithmic Fuctio ( pts) (Clcultor Required) Simplify umeric d lgebric expressios ivolvig l x. Rewrite expressio ivolvig l x s sigle turl logrithm. Rewrite turl logrithmic expressio s the sum/differece of l x with o expoets Solve equtios ivolvig l x f x l x to grph trsformtios. Use the grph of For problems 7-8, solve for x (55) x e 8. (55) lx lx 5 l5 Sectio 56 Derivtives of Logrithmic/Expoetil Fuctios (6 pts) (Clcultor Required) Fid the derivtive of expoetil fuctio. Fid the derivtive of logrithmic fuctio. For problems 9-, fid f ' x. x cos x 9. (56) f x sec e 0. (56) f x e Sequeces d Series Ivestigtios 9

20 . (56) f x lsec x t x. (56) f x l 4x x Sectio 6 Sequeces d Series (8 pts) (Clcultor Required) Alyze ptters i sequeces to determie subsequet terms. Fid the first four terms of sequece give equtio for. Fid expressio for give sequece. Expd series give i sigm ottio. Rewrite give series usig sigm ottio. For problems -4, fid the ext two terms of ech sequece by lyzig the ptter of differeces.. 0,, 4,, 40 4., 8, 8,, 5 For problems 5-6, fid the first four terms of ech sequece ! For problems 7-0, give ech sequece below, fid expressio for. 7., 5, 0, 7, 8. 5,,,, ,,,, , 6, 4, 0, 40 Sequeces d Series Ivestigtios

21 For problems -, write ech series i expded form ! For problems -6, rewrite ech series usig sigm ottio Sectio 6 Arithmetic d Geometric Sequeces (9 pts) (Clcultor Required) Fid the commo differece of rithmetic sequece or the commo rtio of geometric sequece. Express rithmetic or geometric sequece s i terms of. Fid prticulr term of rithmetic or geometric sequece. For problems 7-8: Fid the first four terms of the give sequece. Stte whether the sequece is rithmetic (stte the commo differece), geometric (stte the commo rtio), or either For problems 9-0: Stte whether the sequece is rithmetic (stte the commo differece), geometric (stte the commo rtio), or either. Fid formul for , 40, 0, 0,... 0.,,, Sequeces d Series Ivestigtios 4

22 For problems -, fid the idicted term of ech sequece.. Arithmetic - 9, 5 7,?. Geometric - 5 4, 6, 0? Sectio 6 Fiite Arithmetic d Geometric Series (9 pts) (Clcultor Required) Fid the sum of fiite rithmetic series. Fid the sum of fiite geometric series. For problems -6, fid the sum of ech rithmetic series For problems 7-8, fid the sum of ech geometric series. 7. S 6 :, r 8. S 5 : 6, r Sectio 64 Ifiite Series d Sigm Nottio (9 pts) (Clcultor Required) Fid the sum of ifiite series, if it exists. Apply the ifiite geometric series to modelig scerios. For problems 9-4, fid the sum if the series hs oe. If it does ot, stte tht the series diverges A bll is throw 75 feet ito the ir. Ech time it hits the groud, it rebouds 5 of its previous height. Fid the totl umber of feet the bll trvels before comig to rest. 44. A bll is dropped from height of 00 feet. Ech time it hits the groud, it rebouds 7 of its previous height. 0 Fid the totl umber of feet the bll trvels before comig to rest. 4 Sequeces d Series Ivestigtios

23 Aswers to Selected Exercises Prctice Set 6 Sequeces d Series P. x x e. C. B C 7. l e 0... x x x 9 4., , , , , 7, 0, 8., 4, 8, 6 9.,,, ,, 0,.,,,.,,,.,,, 4., 4, 9, ,,, 6.,,, 7. 0, 0, 6, , 0, 4, ! Prctice Set 6 Arithmetic d Geometric Sequeces P. 6. C. D C 6. C 7. l 8. 7, 9. 4x 4e 0.. t x l0. 5, 7, 9, ; rithmetic; d = 4. 6,, 4, 48; geometric; r = 5.,,, 4 ; either 6.,,, 4 ; either 7., 0,, 0; either 8., 4, 8, 6; geometric; r rithmetic; 4 6. geometric; either; 9. rithmetic; b b 0. geometric; / either; Sequeces d Series Ivestigtios 4

24 Prctice Set 6 Fiite Arithmetic d Geometric Series P. 0. C. B D 7. e 7x e 0. 4x e. x. x 4 x x x e ,8, , Prctice Set 64 Ifiite Series d Sigm Nottio P. 4 x x x x. D. C e cos e 0. e sec e. l x x x x x diverges diverges Prctice Set 65 Assessmet 5 Review P. 8. C. C. B C 6. B cos x e si x. sec x. 5,,, ,,, 9. geometric, r ; 80 l 8. 4 x x x 9. 4e sec e te., , x x ; rithmetic, d = 8. 6, 8, 54, 6 ; geometric, r = 0. either; diverges !,,, diverges 44 Sequeces d Series Ivestigtios

. (b) Evaluate the sum given by. Exercise #1: A sequence is defined by the equation a n 2n

. (b) Evaluate the sum given by. Exercise #1: A sequence is defined by the equation a n 2n Nme: 453 Dte: SEQUENCES ALGEBRA WITH TRIGONOMETRY Sequeces, or ordered list of umbers, re extremely importt i mthemtics, both theoreticl d pplied A sequece is formlly defied s fuctio tht hs s its domi