Chapter 9 Section 9.1 (page 649)

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1 CB_AN.qd // : PM Page Precalculus with Limits, Answers to Section. Chapter Section. (page ) Vocabular Check (page ). infinite sequence. terms. finite. recursivel. factorial. summation notation 7. inde; upper; lower. series. nth partial sum., 7,,,., 7,, 7,.,,,,.,,,,.,,,,. 7.,,.,,,,,, 7,,,, 7.,,, 7, 7.,,,, 7.,,,,.,,,, 7.,.,, 7,,, 7,,.,.,,,,,,, 7.,,,,.,,,,.,,,,..,.,.,.,..,,,,.,,,, a n n a a n n n n! n n.,,,,.,,,, 7.,,,,.,,,,.,,,,.,,,, a n n a n n 7., 7,,,.,,,, a n n a n n.,,.,,,,,, 7.,.,,,,,,,..,,, 7,,,,,,, n 7. n n n n n Copright Houghton Mifflin Compan. All rights reserved c. b. d. a 7. a n n. a. a n n n n. a n n n.. a. a n n n n. n n a n n n n a n n n. a. a 7. a n n n n n n! i i i i.. i i... i 7. i i i 7. i!... i i i i i i i (a) A $., A $., A $., A $., A $., A $., A 7 $7., A $. (b) A $,.. (a) A $., A $., A $., A $., (b) A $. A $,.7 7 i i A $., A $. 7 7 i i i i i

2 CB_AN.qd // : PM Page Precalculus with Limits, Answers to Section. Copright Houghton Mifflin Compan. All rights reserved. (Continued). (a) b n.7n (b) c n.n.n. n. (a) The quadratic model is a better fit. (d) The quadratic model; 7 a n b n c n (b) The number of cases reported fluctuates.. (a) a $., a $., a $7., a $., a $., a $., a $., a 7 $.7, a $., a $., a $7., a $., a $., a $. 7 7 a 7., a., a 7., a., a., a., a., a., a.7 (b) The federal debt is increasing.. $7,. million. True b the Properties of Sums. True. The sums are equal because..,,,,,,,,,,,,,,,,,,,,. Answers will var. 7. $.. $.. Answers will var..,.,,,,..,, 7,,,,,, 7,,,,,,. f. g 7. h,. No inverse. (a) (b) (d) 7. (a) (b) (d). (a) (b) 7 (d) 7. (a) (b) (d)....,7,, 7 7,, 7 7

3 CB_AN.qd // : PM Page Precalculus with Limits, Answers to Section. Copright Houghton Mifflin Compan. All rights reserved. Section. (page ) Vocabular Check (page ). arithmetic; common.. sum of a finite arithmetic sequence a n dn c. Arithmetic sequence, d. Arithmetic sequence, d. Not an arithmetic sequence. Not an arithmetic sequence. Arithmetic sequence, d. Arithmetic sequence, d 7. Not an arithmetic sequence. Arithmetic sequence, d.. Not an arithmetic sequence. Not an arithmetic sequence.,,, 7, Arithmetic sequence, d. 7,,,, Arithmetic sequence, d. 7,,,, Arithmetic sequence, d.,,,, 7 Arithmetic sequence, d.,,,, Not an arithmetic sequence.,,,, Not an arithmetic sequence 7.,,,, Not an arithmetic sequence.,,,, Not an arithmetic sequence. a n n. a n n. a. a n n n n. a n n. a n n. a n n. a n n 7. a n n. a n n. a n n. a n n.,, 7,,., 7, 7,,..,.,.,.,...,.7, 7, 7., 7..,,,,.,,,, 7.,,,,..,.7,, 7.7,..,,, 7, ; d ; a n n.,,,, ; d ; a n n.,,, 7, ; d ; a n n. 7,,,, ; d ; a n n 7.,,,, d a n n ; ;..7,.,.7,.,.7; d.; a n.n b. d. c. a ,., ,., 7., , 7., 7. 7., ,... (a) $, (b) $7,. (a) $, (b) $7,. seats. seats. bricks. bricks 7. meters. 7 feet. (a) a n n (b) $. (a) a n n (b) $7. $7,; answers will var.. $7,. (a) (b) $ Month Monthl $ $ $ $ $ $ pament Unpaid balance $ $ $ $ $ $

4 CB_AN.qd // : PM Page Precalculus with Limits, Answers to Section. Copright Houghton Mifflin Compan. All rights reserved. (Continued). (a) Month Pament $. $7. $. $. Unpaid balance Month 7 Pament $. $7. $. $. Unpaid balance Unpaid balance (b) $. $7 $ $ $. (a) a n n 7, (b) a n.n 7,7; the models are similar., (d) : $, : $,, (e) Answers will var.. (a) Revenue (in millions of dollars) (b) a n 7.n,7. n 7,,,,,, a n ; $, (d) $,. 7. True. Given a and a, d a a and a n a n d.. True b the formula for the sum of a finite arithmetic sequence, S n n a a n.. Answers will var. $7 $ $ $ $7 $ $ $ Month Pament $. $77. $7. $7. a n 7 Year (7 7) n. Add the first term to n times the common difference.. (a) (b) The graph of contains all points on the line. The graph of a n n contains onl points at the positive integers. (d) The slope of the line and the common difference of the arithmetic sequence are equal.. (a),,,, (b) n ; 7 n n n.. S n n; answers will var.. Slope: ;. Slope: ; -intercept: -intercept:, 7. Slope: undefined;. Slope: No -intercept -intercept:,.. 7 a n 7,, z,, z. Answers will var., n 7 7

5 CB_AN.qd // : PM Page Precalculus with Limits, Answers to Section. Section. (page ) Vocabular Check (page ). geometric; common. a n a r n. S n a r n. geometric series r. S a r. a 7. a. c. b. d Copright Houghton Mifflin Compan. All rights reserved., a n. Geometric sequence, r. Geometric sequence, r. Not a geometric sequence. Not a geometric sequence. Geometric sequence, r. Geometric sequence, r. 7. Geometric sequence, r. Geometric sequence, r. Not a geometric sequence. Not a geometric sequence.,,,,.,,,,.,,,,,,,.,,,,.,,,, 7,,,., 7,,,.,,,,,,, 7., e, e, e, e.,,,, 7.,,,.,,,,.,,,, ; r ; n., 7,,, ; r a n n ;. 7,,,, ; r ; a n 7 n.,,,, ; r ; a n n.. 7. a n..,, 7,, ; r ;,,,, ; r ; a n n n ; a n n ;.. a n e n ; e a n n ; a n n a n n, ;... a n n ; 7 a n. n ;.7 a n. n ;..,7.,7, 7.,,.,,. a. a. a , n 7. n 7. 7 n 7. n 7. n 77.. n Undefined. Undefined Horizontal asmptote: Corresponds to the sum of the series. Horizontal asmptote: Corresponds to the sum of the series n n 7 n n n n

6 CB_AN.qd // : PM Page Precalculus with Limits, Answers to Section. (Continued). (a) a n.. n (b) The population is growing at a rate of.% per ear.. million. This value is close to the prediction. (d) 7. (a) $7. (b) $. $. (d) $. (e) $.. (a) $7.7 (b) $7. $7. (d) $7. (e) $7.. $,.. $7.. $.. Answers will var. 7. (a) $,.7 (b) $,.. (a) $,. (b) $,.. (a) $,. (b) $,7.7. (a) $7,. (b) $7,.. Answers will var.. $,.. $. $. $.. $ 7. square inches. S $. million. $,,.. (a). feet (b) seconds. False. A sequence is geometric if the ratios of consecutive terms are the same.. False. a n a r n. Given a real number r between and, as the eponent n increases, r n approaches zero.. Answers will var Does not factor....,.,,.,, 7... Answers will var.,, 7 7 Copright Houghton Mifflin Compan. All rights reserved.

7 CB_AN.qd // : PM Page 7 Precalculus with Limits, Answers to Section. 7 Copright Houghton Mifflin Compan. All rights reserved. Section. (page ) Vocabular Check (page ). mathematical induction. first. arithmetic. second. k k.. k k k. k k. Answers will var.. S n n n. S n n n 7. S n n. S n n. n n S n. S n n n , ,...,,,,, First differences:,,,, Second differences:,,, Linear.,,,,, First differences:,,,, Second differences:,,, Linear.,,,,, 7 First differences:,,,, Second differences:,,, Quadratic.,,,,, First differences:,,, 7, Second differences: 7,,, Neither.,,,,,,,,7, First differences:,,,,,,,,7 Second differences:,,,,,,, Neither.,,,,, First differences:,,,, Second differences:,,, Quadratic 7. a n n n. a n n n 7. a n n n. a n 7 n n. (a).,.,.,.,. (b) A linear model can be used. a n.n.7 a n.n. (d) Part b: a n.; Part c: a n. These are ver similar.. False. P must be proven to be true.. Answers will var.. True. P 7 ma be false.. True. If the second differences are all zero, then the first differences are all the same and the sequence is arithmetic.. False. A sequence with n terms has n second differences (a) Domain: all real numbers ecept (b) Intercept:, Vertical asmptote: Horizontal asmptote: (d) 7. (a) Domain: all real numbers ecept ± (b) Intercept:, Vertical asmptotes: ± Horizontal asmptote: (d) 7. (a) Domain: all real numbers t ecept t (b) t-intercept: 7, Vertical asmptote: t Horizontal asmptote: (d) (, ) (, ) (7, ) t

8 CB_AN.qd // : PM Page Precalculus with Limits, Answers to Section. (Continued) (d) 7. (a) Domain: all real numbers ecept (b) -intercept:, ; -intercept:, Vertical asmptotes: Horizontal asmptote: (, ) (, ) Copright Houghton Mifflin Compan. All rights reserved.

9 CB_AN.qd // : PM Page Precalculus with Limits, Answers to Section. Copright Houghton Mifflin Compan. All rights reserved. Section. (page ) Vocabular Check (page ). binomial coefficients. Binomial Theorem; Pascal s Triangle.. epanding a binomial....., a a a a. a a a a a.... c c d cd d. r r s r s r s r s.. a a b a b 7a b. 7.. a 7a b ab b... n r ; n C r rs 7s ab b t t s t s t s ts s. z z z z 7.. v 7v v v v v , z.,,7 7.,ab.,7,,.. 7.,7,.,7,.. 7.,.,.... t t t 7.. u u u u u. h h, h. h h h, h.. h h,.. i. i. i 7.. i , g is shifted four units to the g f left of f. g 7. h h, g is shifted three units to the right of f. g (a) (b) f t.t.t.t 7.7 (d) f g t.t.t.t 7. g g f

10 CB_AN.qd // : PM Page Precalculus with Limits, Answers to Section. (Continued) (e) f t :. gallons; g t :. gallons; es (f) The trend is for the per capita consumption of bottled water to increase. This ma be due to the increasing concern with contaminants in tap water.. (a) (b) g t.t.t 7. g f. k is the epansion of f.. Answers will var... h g p k = f. True. The coefficients from the Binomial Theorem can be used to find the numbers in Pascal s Triangle.. False. Epanding binomials that represent differences is accurate. The coefficients have alternating signs.. False. The coefficient of the -term is,7, and the coefficient of the -term is,.. The first and last numbers in each row are. Ever other number in each row is formed b adding the two numbers immediatel above the number... n terms f t : 7, g t : The signs of the terms in the epansion of n alternate between positive and negative. g g.. g g. 7.. Copright Houghton Mifflin Compan. All rights reserved.

11 CB_AN.qd // : PM Page Precalculus with Limits, Answers to Section. Copright Houghton Mifflin Compan. All rights reserved. Section. (page ) Vocabular Check (page ). Fundamental Counting Principle. permutation. n P n! r. distinguishable permutations n r!. combinations ,7,.,7, 7. (a) (b) (d). (a) (b) (d).,.,. (a), (b). (a), (b) n or n. n or n.,,.,,,. 7,.,,., ,.....,. ABCD, ABDC, ACBD, ACDB, ADBC, ADCB, BACD, BADC, CABD, CADB, DABC, DACB, BCAD, BDAC, CBAD, CDAB, DBAC, DCAB, BCDA, BDCA, CBDA, CDBA, DBCA, DCBA. ABCD, DBCA, ACBD, DCBA 7.,,,..,,,... AB, AC, AD, AE, AF, BC, BD, BE, BF, CD, CE, CF, DE, DF, EF.,.,.,,. (a) (b). (a) (b) (d) 7. (a) 7 (b). (a) 7 (b).,. (a). (b),..... (a),7, (b) If the jackpot is won, there is onl one winning number. There are,,7 possible winning numbers in the state lotter, which is considerabl less than the possible number of winning Powerball numbers.. (a) Permutation because order matters. (b) Combination because order does not matter. Permutation because order matters. (d) Combination because order does not matter. 7. False. It is an eample of a combination.. True b the definition of the Fundamental Counting Principle.. The are equal. 7. P > C. Changing the order of an of the si elements selected results in a different permutation but the same combination Proof 7. No. For some calculators the number is too great. 7. The smbol n P r denotes the number of was to choose and order r elements out of a collection of n elements. 77. (a) (b) 7. (a) (b) 7. (a) (b). (a) (b)

12 CB_AN.qd // : PM Page Precalculus with Limits, Answers to Section.7 Copright Houghton Mifflin Compan. All rights reserved. Section.7 (page 7) Vocabular Check (page 7). eperiment; outcomes. sample space. probabilit. impossible; certain. mutuall eclusive. independent 7. complement. (a) iii (b) i iv (d) ii. H,, H,, H,, H,, H,, H,, T,, T,, T,, T,, T,, T,.,,...,. ABC, ACB, BAC, BCA, CAB, CBA. red, red, red, blue, red, ellow, blue, blue, blue, ellow. AB, AC, AD, AE, BC, BD, BE, CD, CE, DE. SSS, SSF, SFS, SFF, FSS, FSF, FFS, FFF (for A ) (a) % (b).%.%. (a) % (b) % %. (a) (b). (a) (b) 7. (a) (b) (a) (b). P Talor wins P Moore wins P Jenkins wins %. (a) (b). (a) (b). (a) (b). (a).7 (b).. (a) (b). 7. (a) (b). (a) (b) (d) (a).7 (b)... (a). (b)... (a) 7 (b).. (a) 7 (b) (d) (e) (f) The probabilities are slightl better in European roulette. 7. (a) (b) Approimations will var. 7. True. Two events are independent if the occurrence of one has no effect on the occurrence of the other.. False. The complement of the event is to roll a number greater than or equal to, and its probabilit is.. (a) As ou consider successive people with distinct birthdas, the probabilities must decrease to take into account the birth dates alread used. Because the birth dates of people are independent events, multipl the respective probabilities of distinct birthdas. (b) Answers will var. (d) Q n is the probabilit that the birthdas are not distinct, which is equivalent to at least two people having the same birthda. (e) n (f). Meteorological records indicate that over an etended period of time with similar weather conditions it will rain % of the time. ± 7. No real solution.., ±., ±.. ± P n Q n

13 CB_AN.qd // : PM Page Precalculus with Limits, Answers to Review Eercises Copright Houghton Mifflin Compan. All rights reserved. Review Eercises (page 7) 7.,,,,.,,, 7,. 7,,,,.,,,,. a n n. a n n 7. a. a n n n n n k k. k.... k k. (a) A $,.7, A $,.7, A $,., A $,., A $,7., A $,.7, A 7 $,7., A $,., A $,., A $,7. (b). (a) A $,. a 7., a 7., a 7., a 7 7., a., a.7, a, a.7, a. 7. Arithmetic sequence, d. Not an arithmetic sequence. Arithmetic sequence, d. Arithmetic sequence, d., 7,,,.,,,,.,,,, 7..,.,.,.,.. a n n. a n n 7. a n n. a n n. a n 7n 7. a n n.....,. 7. (a) $, (b) $,. 7 bales. Geometric sequence, r. Geometric sequence, r. Geometric sequence, r. Not a geometric sequence.,,,,.,,,,.,,, or,,,,,.,,,, 7 or,,,, 7 7. a n n ;.. a n n ;.. a n. n ;.. a n. n ; (a) a t,.7 t (b) $,. 7. $,.7; $,7. 7. Answers will var.. S n n n 7. S n n n. S n n. S n 7....,.,,,, First differences:,,, Second differences:,, Linear., 7,,, First differences:,,, Second differences:,, Quadratic.,,,, First differences:,,, Second differences:,, Linear.,,,, First differences:,,, Second differences:,, Neither a a b a b 7a b ab b i. i..., ,7,.. 7, 7... (a) % (b) %. (a).% (b) % 7.%..... True. n! n! n n n! n!. True b Properties of Sums 7. True b Properties of Sums. True. The sums are equal because n n 7.

14 CB_AN.qd // : PM Page Precalculus with Limits, Answers to Review Eercises (Continued). False. When r equals or, then the results are the same.. The set of natural numbers. In the sequence in part (a), the odd-numbered terms are negative, whereas in the sequence in part (b), the evennumbered terms are negative.. (a) Arithmetic. There is a constant difference between consecutive terms. (b) Geometric. Each term is a constant multiple of the preceding term. In this case, the common ratio is greater than.. Each term of the sequence is defined in terms of preceding terms.. Increasing powers of real numbers between and approach zero.. d. a 7. b. c.,,,, 7. p ; Closed interval Copright Houghton Mifflin Compan. All rights reserved.

15 CB_AN.qd // : PM Page Precalculus with Limits, Answers to Chapter Test Chapter Test (page 7).,,,, 7. a n n n!.,, 7;. a n.n..,,,,., 7... Answers will var...,. (a) 7 (b),. (a) (b) 7,7., % Copright Houghton Mifflin Compan. All rights reserved.

16 CB_AN.qd // : PM Page Precalculus with Limits, Answers to Cumulative Test for Chapters 7 Cumulative Test for Chapters 7 (page 7).,,,.,.,,.,,.. 7. (, ) (, ) (, ) (, ) Minimum at, : z 7 Maimum at, : z. $.7 miture: pounds; $. miture: pounds...,, Gm shoes: $. million Jogging shoes: $. million Walking shoes: $7.7 million.,.,,.. n!. a n, 7,,, n.. (a). (b) a n.n..,,,, 7.. Answers will var.. z z z z ,., Copright Houghton Mifflin Compan. All rights reserved.

17 CB_AN.qd // : PM Page 7 Precalculus with Limits, Answers to Problem Solving 7 Copright Houghton Mifflin Compan. All rights reserved. Problem Solving (page 7).,.,.,.,.,.,... n approaches.. (a) (b) (d). (a) n, a n (b) If n is odd, a n, and if n is even, a n. n, a n.... (d) It is not possible to find the value of a n as n approaches infinit.. (a) Arithmetic sequence, difference d (b) Arithmetic sequence, difference dc Not an arithmetic sequence. (a),, 7,,,,, 7 a n n (b) To obtain the arithmetic sequence, find the differences of consecutive terms of the sequence of perfect cubes. Then find the differences of consecutive terms of this sequence.,,,,,, a n n (d) To obtain the arithmetic sequence, find the third sequence obtained b taking differences of consecutive terms in consecutive sequences. (e),,,,, a n n. s d a r This represents the total distance Achilles ran. s t a r This represents the total amount of time Achilles ran. 7. s n n a n s n. (a) 7,,,, 7,,,,, (b) Eventuall the terms repeat:,, if a is a positive integer and, if a is a negative integer.. Answers will var.. (a) P n is true for integers n. (b) P n is true for integers n. P, P, and P are true. (d) is true for an positive integer n. P n. (a) Answers will var. (b) 7,7. (a) marbles (b) to 7; 7 to odds in favors of E P E odds in favor of E (d) Odds in favor of event E P E P E... (a) $.7 (b)., turns