10-3 Geometric Sequences and Series

Transcription

1 1 2 Determine the common ratio, and find the next three terms of each geometric sequence = 2 = 2 The common ratio is 2 Multiply the third term by 2 to find the fourth term, and so on 1( 2) = 2 2( 2) = 4 4( 2) = 8 Therefore, the next three terms are 2, 4, and 8 The common ratio is by Multiply the third term to find the fourth term, and so on Therefore, the next three terms are,, and 3 05, 075, 1125, = = 15 The common ratio is 15 Multiply the third term by 15 to find the fourth term, and so on 1125(15) = (15) = (15) = Therefore, the next three terms are 16875, , and , 20, 50, 20 8 or or 25 The common ratio is 25 Multiply the third term by 25 to find the fourth term, and so on 50(25) = (25) = (25) = Therefore, the next three terms are 125, 3125, and x, 10x, 50x, 10x 2x = 5 50x 10x = 5 The common ratio is 5 Multiply the third term by 5 to find the fourth term, and so on 5(50x) = 250x 5(250x) =1250x 5(1250x) =6250x Therefore, the next three terms are 250x, 1250x, and 6250x esolutions Manual - Powered by Cognero Page 1

2 6 64x, 16x, 4x, 8 9 y, y, 81 9y, 16x 64x = 4x 16x = The common ratio is Multiply the third term by to find the fourth term, and so on The common ratio is 3 Multiply the third term by 3 to find the fourth term, and so on 3( 81 9y) = y 3( y) = y 3( y) = y Therefore, the next three terms are y, y, and y Therefore, the next three terms are x, x, and x 7 x + 5, 3x + 15, 9x + 45, The common ratio is 3 Multiply the third term by 3 to find the fourth term, and so on 3(9x + 45) = 27x (27x +135) = 81x (81x + 405) = 243x Therefore, the next three terms are 27x +135, 81x + 405, and 243x esolutions Manual - Powered by Cognero Page 2

3 9 GEOMETRY Consider a sequence of circles with diameters that form a geometric sequence: d 1, d 2, d 3, d 4, d 5 Write an explicit formula and a recursive formula for finding the nth term of each geometric sequence 10 36, 12, 4, = a Show that the sequence of circumferences of the circles is also geometric Identify r b Show that the sequence of areas of the circles is also geometric Identify the common ratio a Sample answer: The circumference of a circle is given by C = πd So, the sequence of circumferences of the circles is πd 1, πd 2, πd 3, πd 4, πd 5 Find the common ratio 4 12 = For an explicit formula, substitute a 1 = 36 and r = in the nth term formula For a recursive formula, state the first term a 1 Then indicate that the next term is the product of the first term a n 1 and r a 1 = 36, a n = b Sample answer: The area of a circle is given by C = πr 2 or of the circles is So, the sequence of areas Find the common ratio esolutions Manual - Powered by Cognero Page 3

4 11 64, 16, 4, = 4 16 = For an explicit formula, substitute a 1 = 64 and r = 13 4, 12, 36, 12 4 = = 3 For an explicit formula, substitute a 1 = 4 and r = 3 in the nth term formula in the nth term formula For a recursive formula, state the first term a 1 Then indicate that the next term is the product of the first term a n 1 and r a 1 = 64, a n = 12 2, 10, 50, 10 2 = = 5 For an explicit formula, substitute a 1 = 2 and r = 5 in the nth term formula For a recursive formula, state the first term a 1 Then indicate that the next term is the product of the first term a n 1 and r a 1 = 4, a n = 14 4, 8, 16, 8 4 = = 2 For an explicit formula, substitute a 1 = 4 and r = 2 in the nth term formula For a recursive formula, state the first term a 1 Then indicate that the next term is the product of the first term a n 1 and r a 1 = 4, a n = For a recursive formula, state the first term a 1 Then indicate that the next term is the product of the first term a n 1 and r a 1 = 2, a n = esolutions Manual - Powered by Cognero Page 4

5 15 20, 30, 45, = = 15 For an explicit formula, substitute a 1 = 20 and r = 15 in the nth term formula 17,,, = 2 = 2 For an explicit formula, substitute a 1 = and r = For a recursive formula, state the first term a 1 Then indicate that the next term is the product of the first term a n 1 and r a 1 = 20, a n = 16 15, 5,, 2 in the nth term formula For a recursive formula, state the first term a 1 Then indicate that the next term is the product of the first term a n 1 and r a 1 =, a n = 5 15 = 5 = For an explicit formula, substitute a 1 = 15 and r = in the nth term formula For a recursive formula, state the first term a 1 Then indicate that the next term is the product of the first term a n 1 and r a 1 = 15, a n = esolutions Manual - Powered by Cognero Page 5

6 18 CHAIN Melina receives a chain that she forwards to 7 of her friends Each of her friends forwards it to 7 of their friends 19 BIOLOGY A certain bacteria divides every 15 minutes to produce two complete bacteria a If an initial colony contains a population of b 0 bacteria, write an equation that will determine the number of bacteria b t present after t hours b Suppose a Petri dish contains 12 bacteria Use the equation found in part a to determine the number of bacteria present 4 hours later a Write an explicit formula for the pattern b How many will receive the after 6 forwards? Melina receives a chain , forwards it to 7 friends, and each friend forwards it to 7 friends Therefore, a 1 = 1, a 2 = 7, and a 3 = 49 The common ratio is 7 For an explicit formula, substitute a 1 = 1 and r = 7 in the nth term formula b Use the explicit formula you found in part a to find a 6 a Initially, there is 1 bacterium After 15 minutes, there will be 2 bacteria, after 30 minutes there will be 4 bacteria, after 45 minutes there will be 8 bacteria, and after 1 hour there will be 16 bacteria So, in terms of hours, b 0 = 1 and b 1 = 16 Find the common ratio 16 1 = 16 Write an explicit formula using r = 16 b Substitute b 0 = 12 and t = 4 into the equation you found in part a Therefore, after 6 forwards 16,807 people will have received the esolutions Manual - Powered by Cognero Page 6

7 Find the specified term for each geometric sequence or sequence with the given characteristics 20 a 9 for 60, 30, 15, = = Use the formula for the nth term of a geometric sequence to find a 9 22 a 5 for 3, 1,, 1 3 = 1 = Use the formula for the nth term of a geometric sequence to find a 5 23 a 6 for 540, 90, 15, 21 a 4 for 7, 14, 28, 14 7 = = 2 Use the formula for the nth term of a geometric sequence to find a = = Use the formula for the nth term of a geometric sequence to find a 6 esolutions Manual - Powered by Cognero Page 7

8 24 a 7 if a 3 = 24 and r = 05 Use the values of a 3 and r to find a 2 25 a 6 if a 3 = 32 and r = 05 Use the values of a 3 and r to find a 2 Next, use the values of a 2 and r to find a 1 Next, use the values of a 2 and r to find a 1 Use the formula for the nth term of a geometric sequence to find a 7 Use the formula for the nth term of a geometric sequence to find a 6 Another method would be to consider that the 7th term is 4 terms from the 3rd term Therefore, multiply the 3rd term by r 4 Another method would be to consider that the 6th term is 3 terms from the 3rd term Therefore, multiply the 3rd term by r 3 26 a 6 if a 1 = 16,807 and r = Use the formula for the nth term of a geometric sequence to find a 6 esolutions Manual - Powered by Cognero Page 8

9 27 a 8 if a 1 = 4096 and r = Use the formula for the nth term of a geometric sequence to find a 8 29 Find the sixth term of a geometric sequence with a first term of 9 and a common ratio of 2 Use the formula for the nth term of a geometric sequence to find the a 6 30 If r = 4 and a 8 = 100, what is the first term of the geometric sequence? 28 ACCOUNTING Julian Rockman is an accountant for a small company On January 1, 2009, the company purchased $50,000 worth of computers, printers, scanners, and hardware Because this equipment is a company asset, Mr Rockman needs to determine how much the computer equipment is presently worth He estimates that the computer equipment depreciates at a rate of 45% per year What value should Mr Rockman assign the equipment in his 2014 year-end accounting report? The equipment is originally worth $50,000, so a 1 = 50,000 Because the equipment depreciates at a rate of 45% per year, the value of the equipment on a given year will be 100% 45% or 55% of the value the previous year So, r = 055 The first term a 1 corresponds to the year 2009, so the year 2014 corresponds to a 6 Use the formula for the nth term of a geometric sequence to find the a 6 Substitute a 8 = 100, r = 4, and n = 8 into the formula for the nth term of a geometric sequence to find the a 1 31 X GAMES Refer to the beginning of the lesson The X Games netted approximately $40 million in revenue in 2002 If the X Games continue to generate 13% more revenue each year, how much revenue will the X Games generate in 2020? The X Games netted about $40,000,000 in 2002, so a 1 = $40,000,000 Because the games generate 13% or 013 more revenue each year, the amount of revenue generated on a given year will be 113 times the revenue from the previous year So, r = 113 The first term a 1 corresponds to the year 2002, so the year 2020 corresponds to a 19 Use the formula for the nth term of a geometric sequence to find the a 19 Therefore, the value of the equipment in 2014 is about $ Therefore, the X Games will generate about $36097 million in 2020 esolutions Manual - Powered by Cognero Page 9

10 Find the indicated geometric means for each pair of nonconsecutive terms 32 4 and 256; 2 means The sequence will resemble 4,?,?, and 81; 3 means The sequence will resemble 256,?,?,?, 81 Note that a 1 = 256, n = 5, and a 5 = 81 Find the common ratio using the nth term for a geometric sequence formula Note that a 1 = 4, n = 4, and a 4 = 256 Find the common ratio using nth term for a geometric sequence formula The common ratio is 4 Use r to find the geometric means 4(4) = 16 16(4) = 64 Therefore, a sequence with two geometric means between 4 and 256 is 4, 16, 64, 256 The common ratio is geometric means r = Use r to find the r = Therefore, a sequence with three geometric means between 256 and 81 is 256, 192, 144, 108, 81 or 256, 192, 144, 108, 81 esolutions Manual - Powered by Cognero Page 10

11 34 and 7; 1 mean The sequence will resemble that a 1 =,?, 7 Note, n = 3, and a 3 = 7 Find the common ratio using the nth term for a geometric sequence formula 35 2 and 54; 2 means The sequence will resemble 2,?,?, 54 Note that a 1 = 2, n = 4, and a 4 = 54 Find the common ratio using nth term for a geometric sequence formula The common ratio is geometric means Use r to find the The common ratio is 3 Use r to find the geometric means 2( 3) = 6 6( 3) = 18 Therefore, a sequence with two geometric means between 2 and 54 is 2, 6, 18, and 27; 2 means Therefore, a sequence with one geometric mean between and 7 is, 2, 7 or, 2, 7 The sequence will resemble 1,?,?, 27 Note that a 1 = 1, n = 4, and a 4 = 27 Find the common ratio using nth term for a geometric sequence formula The common ratio is 3 Use r to find the geometric means 1(3) = 3 3(3) = 9 Therefore, a sequence with two geometric means between 1 and 27 is 1, 3, 9, 27 esolutions Manual - Powered by Cognero Page 11

12 37 48 and 750; 2 means The sequence will resemble 48,?,?, 750 Note that a 1 = 48, n = 4, and a 4 = 48 Find the common ratio using nth term for a geometric sequence formula 38 i and 1; 4 means The sequence will resemble i,?,?,?,?, 1 Note that a 1 = i, n = 6, and a 6 = 1 Find the common ratio using nth term for a geometric sequence formula The common ratio is geometric means = 120 = 300 Use r to find the The common ratio is i Use r to find the geometric means i(i) = 1 1(i) = i i(i) = 1 1(i) = i Therefore, a sequence with two geometric means between i and 1 is i, 1, i, 1, i, 1 Therefore, a sequence with two geometric means between 48 and 750 is 48, 120, 300, 750 esolutions Manual - Powered by Cognero Page 12

13 39 t 8 and t 7 ; 4 means The sequence will resemble t 8,?,?,?,?, t 7 Note that a 1 = t 8, n = 6, and a 6 = t 7 Find the common ratio using nth term for a geometric sequence formula Find the sum of each geometric series described 40 first six terms of = = 3 The common ratio is 3 Use Formula 1 for the sum of a finite geometric series The common ratio is Use r to find the geometric means = t 5 = t 2 = t 1 = t 4 41 first nine terms of 05 + ( 1) or or 2 The common ratio is 2 Use Formula 1 for the sum of a finite geometric series Therefore, a sequence with two geometric means between t 8 and t 7 is t 8, t 5, t 2, t 1, t 4, t 7 esolutions Manual - Powered by Cognero Page 13

14 42 first eight terms of or 6 2 or The common ratio is Use Formula 1 for the sum of a finite geometric series 44 [Contains update not in print edition] first n terms of a 1 = 5, a n = 1,310,720, r = 4 [Solution for updated problem] Use Formula 2 for the nth partial sum of a geometric series 45 first n terms of a 1 = 3, a n = 46,875, r = 5 Use Formula 2 for the nth partial sum of a geometric series 43 first n terms of a 1 = 4, a n = 2000, r = 3 Use Formula 2 for the nth partial sum of a geometric series esolutions Manual - Powered by Cognero Page 14

15 46 first n terms of a 1 = 8, a n = 256, r = 2 Use Formula 2 for the nth partial sum of a geometric series Find each sum 48 Find n, a 1, and r Substitute n = 6, a 1 = 5, and r = 2 into the formula for the sum of a finite geometric series 47 first n terms of a 1 = 36, a n = 972, r = 7 Use Formula 2 for the nth partial sum of a geometric series 49 Find n, a 1, and r Substitute n = 5, a 1 = 4, and r = 3 into the formula for the sum of a finite geometric series esolutions Manual - Powered by Cognero Page 15

16 50 52 Find n, a 1, and r Find n, a 1, and r Substitute n = 5, a 1 = 1, and r = 3 into the formula for the sum of a finite geometric series Substitute n = 6, a 1 = 100, and r = into the formula for the sum of a finite geometric series 51 Find n, a 1, and r Substitute n = 6, a 1 = 2, and r = 14 into the formula for the sum of a finite geometric series esolutions Manual - Powered by Cognero Page 16

17 53 54 Find n, a 1, and r Find n, a 1, and r Substitute n = 9, a 1 =, and r = 3 into the formula for the sum of a finite geometric series Substitute n = 7, a 1 = 144, and r = into the formula for the sum of a finite geometric series esolutions Manual - Powered by Cognero Page 17

18 55 If possible, find the sum of each infinite geometric series Find n, a 1, and r = Substitute n = 20, a 1 = 3, and r = 2 into the formula for the sum of a finite geometric series = The common ratio r is < 1 Therefore, this infinite geometric series has a sum Use the formula for the sum of an infinite geometric series Therefore, the sum of the series is = 2 = 2 The common ratio r is > 1 Therefore, this infinite geometric series has no sum esolutions Manual - Powered by Cognero Page 18

19 ( 27) = = 15 The common ratio r is > 1 Therefore, this infinite geometric series has no sum ( 72) = = The common ratio is < 1 Therefore, this infinite geometric series has a sum Find a 1 Use the formula for the sum of an infinite geometric series to find the sum The common ratio r is < 1 Therefore, this infinite geometric series has a sum Use the formula for the sum of an infinite geometric series Therefore, the sum of the series is Therefore, the sum of the series is 75 esolutions Manual - Powered by Cognero Page 19

20 61 62 The common ratio is < 1 Therefore, this infinite The common ratio is < 1 Therefore, this infinite geometric series has a sum Find a 1 geometric series has a sum Find a 1 Use the formula for the sum of an infinite geometric series to find the sum Use the formula for the sum of an infinite geometric series to find the sum Therefore, the sum of the series is 100 Therefore, the sum of the series is esolutions Manual - Powered by Cognero Page 20

21 63 The common ratio is < 1 Therefore, this infinite geometric series has a sum Find a 1 Use the formula for the sum of an infinite geometric series to find the sum 64 BUNGEE JUMPING A bungee jumper falls 35 meters before his cord causes him to spring back up He rebounds of the distance after each fall a Find the first five terms of the infinite sequence representing the vertical distance traveled by the bungee jumper Include each drop and rebound distance as separate terms b What is the total vertical distance the jumper travels before coming to rest? (Hint: Rewrite the infinite sequence suggested by in part a as two infinite geometric sequences) a The bungee jumper will fall 35 meters, spring back up back up (35) or 14 meters, fall 14 meters, spring (14) or 56 meters, fall 56 meters, spring back up (56) or 224, and so on Therefore, the sum of the series is 20 Therefore, the first five terms of the infinite sequence that represents the vertical distance traveled by the bungee jumper are 35, 14, 14, 56, and 56 b The series that corresponds to the infinite sequence 35, 14, 14, 56, 56, can be written as the sum of the two infinite geometric series: one series that represents the distance traveled when falling and one series that represents the distance traveled when springing back up Series Series Find the sum of each series Therefore, the total vertical distance that the jumper travels is or about 82 m esolutions Manual - Powered by Cognero Page 21

22 Find the missing quantity for the geometric sequence with the given characteristics 65 Find a 1 if S 12 = 1365 and r = 2 Substitute S 12 = 1365, n = 12, and r = 2 into the formula for the sum of a finite geometric series 67 Find r if a 1 = 012, S n = 59052, and a n = Substitute S n = 59052, a 1 = 012, and a n = into the formula for the nth partial sum of an infinite geometric series 68 Find n for if S n = If S 6 = , a 1 = 100, r = 05, find a 6 Substitute S 6 = , a 1 = 100, and r = 05 into the formula for the nth partial sum of an infinite geometric series The common ratio is 2 Substitute S n = 615, a 1 = 41, and r = 2 into the formula for the nth partial sum of an infinite geometric series Substitute a n = 328, a 1 = 41, and r = 2 into the formula for the nth term of a geometric sequence to find n esolutions Manual - Powered by Cognero Page 22

23 69 If , S n = 23784, find a n Find the common ratio = = Find a 1 if S n = 468, a n = 375, and r = 5 Substitute S n = 468, a n = 375, and r = 5 into the formula for the nth partial sum of an infinite geometric series Substitute S n = 23784, a 1 = 15, and r = 12 into the formula for the nth partial sum of an infinite geometric series 70 If r = 04, S 5 = 14432, and a 1 = 200, find a 5 Substitute S 5 = 14432, a 1 = 200, and r = 04 into the formula for the nth partial sum of an infinite geometric series esolutions Manual - Powered by Cognero Page 23

24 72 If S n =, + + +, find n Find the common ratio = = Substitute S n =, a 1 =, and r = into the formula for the nth partial sum of an infinite geometric series 73 LOANS Marc is making monthly payments on a loan Suppose instead of the same monthly payment, the bank requires a low initial payment that grows at an exponential rate each month The total cost of the loan is represented by a What is Marc s initial payment and at what rate is this payment increasing? b If the sum of Marc s payments at the end of the loan is $7052, how many payments did Marc make? a The first term of the series represented by this sigma notation is a 1 = 5(11) 1 1 or 5 This represents the first month payment on the loan of $5 The rate at which the payment is increasing r is the base of the exponential function, 11 b Write and solve an equation using Formula 1 for the sum of finite geometric series to find k Because a n = and the third term of the sequence is, n = 3 Therefore, Marc made 52 payments esolutions Manual - Powered by Cognero Page 24

25 10-3 Geometric Sequences and Series Therefore, Marc made 52 payments Find the common ratio for the geometric sequence with the given terms 74 a 3 = 12, a 6 = 1875 The 6th term is 3 terms away from the 3rd term 75 a 2 = 6, a 7 = 192 The 7th term is 5 terms away from the 2nd term 78 ADVERTISING Word-of-mouth advertising can be an effective form of marketing, or it can be very harmful Consider a new restaurant that serves 27 customers on its opening night a Of the 27 customers, 25 found the experience enjoyable and each told 3 friends over the next month This group each told 3 friends over the next month, and so on, for 6 months Assuming that no one heard twice, how many people have had a positive experience or heard positive reviews of the restaurant? b Suppose the 2 unhappy customers each told 6 friends over the next month about the experience This group then each told 6 friends, and so on, for 6 months Assuming that no one heard a review twice, how many people have had a negative experience or have heard a negative review? a In the sequence representing customers that had a positive experience or heard a positive review, a 1 = 25 and r = 3 Find the sum of the first six terms 76 a 4 = 28, a 6 = 1372 The 6th term is 2 terms away from the 4th term Therefore, 9100 people had a positive experience or heard a positive review in the first 6 months 77 a 5 = 6, a 8 = 0048 b In the sequence representing customers that had a negative experience or heard a negative review, a 1 = 2 and r = 6 Find the sum of the first six terms The 8th term is 3 terms away from the 5th term Therefore, 18,662 people had a negative experience or heard a negative review in the first 6 months 78 ADVERTISING Word-of-mouth advertising can be an effective form of marketing, or it can be very harmful Consider a new restaurant that serves 27 customers on its opening night a Of the 27 customers, 25 found the experience esolutions Manual - Powered by Cognero enjoyable and each told 3 friends over the next Write the first 3 terms of the infinite geometric series with the given characteristics 79 S = 12, r = Page 25

26 Write the first 3 terms of the infinite geometric series with the given characteristics 79 S = 12, r = 80 S = 25, r = 02 Substitute S = 25 and r = 02 into the formula for the sum of an infinite geometric series to find a 1 Substitute S = 12 and r = into the formula for the sum of an infinite geometric series to find a 1 Use r = 02 to find a 2 and a 3 20(02) = 4 4(02) = Therefore, the first three terms of the sequence are Use r = to find a 2 and a 3 20, 4, and 81 S = 448, a 1 = 56 Therefore, the first three terms of the sequence are 6, 3, and Substitute S = 25 and a 1 = 56 into the formula for the sum of an infinite geometric series to find r Use r = 025 to find a 2 and a 3 56( 025) = 14 14( 025) = 35 Therefore, the first three terms of the sequence are 56, 14, and 35 esolutions Manual - Powered by Cognero Page 26

27 82 S =, a 1 = Substitute S = and a 1 = into the formula for 83 S = 60, r = 04 Substitute S = 60 and r = 04 into the formula for the sum of an infinite geometric series to find a 1 the sum of an infinite geometric series to find r Use r = 04 to find a 2 and a 3 36(04) = (04) = 576 Therefore, the first three terms of the sequence are 36, 144, and 576 Use r = to find a 2 and a 3 Therefore, the first three terms of the sequence are 84 S = 12625, a 1 = 505 Substitute S = and a 1 = 505 into the formula for the sum of an infinite geometric series to find r,, and Use r = 06 to find a 2 and a 3 505(06) = (06) = 1818 Therefore, the first three terms of the sequence are 505, 303, and 1818 esolutions Manual - Powered by Cognero Page 27

28 85 S = 115, a 1 = 138 Substitute S = 115 and a 1 = 138 into the formula for the sum of an infinite geometric series to find r 86 S =, r = Substitute S = and r = into the formula for the sum of an infinite geometric series to find a 1 Use r = 02 to find a 2 and a 3 138( 02) = ( 02) = 552 Therefore, the first three terms of the sequence are 138, 276, and 552 Use r = to find a 2 and a 3 Therefore, the first three terms of the sequence are,, and esolutions Manual - Powered by Cognero Page 28

29 87 Find a 1 Determine whether each sequence is arithmetic, geometric, or neither Then find the next three terms of the sequence Use r = to find a 2 and a 3 Therefore, the first three terms of the sequence are 12, 3, and 88 Find a 1 Use r = to find a 2 and a 3 Therefore, the first three terms of the sequence are,, and esolutions Manual - Powered by Cognero Page 29

30 89,,,, First, determine whether there is a common difference or common ratio = = 90,, 4,, First, determine whether there is a common difference or common ratio = 4 = = = = 4 = Because there is no common difference or common ratio, the sequence is neither arithmetic nor geometric Notice that there appears to be a pattern in which the numerator of each fraction is increasing by 1 and the denominator is increasing by 2 for each successive term a 1 = = = Because there is a common difference of, this sequence is arithmetic Find the next three terms of the sequence + = + = + = 3 Therefore, the next three terms are,, and 3 = Find the next three terms of the sequence = = = Therefore, the next three terms are,, and esolutions Manual - Powered by Cognero Page 30

31 91 12, 24, 36, 48, First, determine whether there is a common difference or common ratio = = = = Because there is a common difference of 12, this sequence is arithmetic Find the next three terms of the sequence = = = 84 Therefore, the next three terms are 60, 72, and , 96, 72, 54, First, determine whether there is a common difference or common ratio = = = = 075 Because there is a common ratio of 075, this sequence is geometric Find the next three terms of the sequence 54(075) = (075) = (075) = Therefore, the next three terms are 405, 30375, and k, 49k, 64k, 81k, First, determine whether there is a common difference or common ratio 49k 36k = 13k 64k 49k = 25k 49k 36k = 64k 49k = Because there is no common difference or common ratio, the sequence is neither arithmetic nor geometric Notice that there appears to be a pattern in which the first term is 6 2 k, the second term is 7 2 k, the third term is 8 2 k, and the fourth term is 9 2 k Find the next three terms of the sequence 10 2 k = 100k 11 2 k = 121k 12 2 k = 144k Therefore, the next three terms are 100k, 121k, and 144k 94 72y, 91y, 11y, 129y, First, determine whether there is a common difference or common ratio 91y 72y = 19y 11y 91y = 19y 91y 72y = about y 91y = about 121 Because there is a common difference of 19y, this sequence is arithmetic Find the next three terms of the sequence 129y + 19y = 148y 148y + 19y = 167y 167y + 19y = 186y Therefore, the next three terms are 148y, 167y, and 186y esolutions Manual - Powered by Cognero Page 31

32 95 3, 15, 15, 75, First, determine whether there is a common difference or common ratio 15 3 = about = about = = Because there is a common ratio of, this sequence is geometric Find the next three terms of the sequence Therefore, the next three terms are 75, 375, and , 2, 2, 2, First, determine whether there is a common difference or common ratio 2 2 = about = about = 2 2 = Because there is no common difference or common ratio, the sequence is neither arithmetic nor geometric Notice that there appears to be a pattern in which the radicand is increasing by 3 for each successive term Find the next three terms of the sequence 2 = 2 2 = 2 2 = 2 Therefore, the next three terms are 2, 2, and 2 esolutions Manual - Powered by Cognero Page 32

33 Write each geometric series in sigma notation Find the common ratio 12 3 = = 4 Next, determine the upper bound a 4 = 48(4) = 192 a 5 = 192(4) = 768 a 6 = 768(4) = 3072 Write an explicit formula for the sequence Therefore, in sigma notation the series can be written as Find the common ratio = = 17 Next, determine the upper bound a 4 = 1445(17) = a 5 = 24565(17) = Write an explicit formula for the sequence Therefore, in sigma notation the series can be written as Find the common ratio 18 9 = = 2 Next, determine the upper bound a 4 = 36(2) = 72 a 5 = 72(2) = 144 a 6 = 144(2) = 288 a 7 = 288(2) = 576 a 8 = 576(2) = 1152 Write an explicit formula for the sequence Therefore, in sigma notation the series can be written as esolutions Manual - Powered by Cognero Page 33

34 100 Find the common ratio = 2 = 2 Next, determine the upper bound a 4 = a 5 = 1( 2) = 2 a 6 = 2( 2) = 4 a 7 = 4( 2) = 8 Write an explicit formula for the sequence Find the common ratio 1 02 = = 5 Next, determine the upper bound a 4 = 5( 5) = 25 a 5 = 25( 5) = 125 a 6 = 125( 5) = 625 Write an explicit formula for the sequence Therefore, in sigma notation the series 02 + ( 1) ( 625) can be written as Therefore, in sigma notation the series can be written as 102 HORSES For each of the first few months after a horse is born, the amount of weight that it gains is about 120% of the previous month s weight gain In the first month, a horse has gained 30 pounds a Write a geometric series in sigma notation that can be used to model the horse's weight gain for the first five months b About how much weight did the horse gain in the fourth month? c If the horse weighed 150 pounds at birth, about how much did it weigh after 5 months? d Will the horse continue to grow at this rate indefinitely? Explain a The amount of weight that a horse gains is 120% of the previous month's weight gain, so r = 12 Because the horse gained 30 pounds in the first month, a 1 = 30 Write an explicit formula for the horse's weight gain The lower bound is 1 and the upper bound is 5 Therefore, in sigma notation the series representing the horse's weight gain for the first five months can esolutions Manual - Powered by Cognero Page 34

35 the horse's weight gain for the first five months can be written as find the amount of medicine at t = 6, the time of the second dose b Use the explicit formula you found in part a to find a 4 Therefore, in the fourth month the horse gained 5184 pounds c First, find the sum of the series Therefore, after 6 hours will be left in the patient's system of the original dose b Immediately following the first dose, the amount of medicine in the patient's system is d, so a 1 = d Immediately following the second dose, the amount of medicine in the patient's system is the sum of the amount of medicine left from the first dose and the amount taken during the second dose, so a 2 = d Because the horse weighed 150 pounds at birth, after 5 months the horse weighed or about pounds d Sample answer: No; the horse cannot continue to grow at this rate indefinitely because its body will eventually stop growing 103 MEDICINE A newly developed and researched medicine has a half-life of about 15 hours after it is administered The medicine is given to patients in doses of d milligrams every 6 hours a What fraction of the first dose will be left in the patient s system when the second dose is taken? b Find the first four terms of the sequence that represents the amount of medicine in a patient's system after the first 4 doses c Write a recursive formula that can be used to determine the amount of medicine in the patient's system after the nth dose a Let d represent the first dose at time t = 0 After 15 hours or t = 15 the amount of medicine left in + d or d Find a 2 and a 3, the amount of medicine that is left immediately after taking the third and fourth doses, which occur at times t = 12 and t = 18, respectively the patient's system will be d Use this pattern to Therefore, the first four terms of the sequence are esolutions Manual - Powered by Cognero Page 35

36 d, d, d, and d c Notice that a pattern forms when the first four terms are rewritten as shown below c ANALYTICAL For each graph in part a, describe the values of S n as n d GRAPHICAL Graph S n = for r = 12, 25, and 4 on the same graph e ANALYTICAL For each graph in part d, describe the values of S n as n f ANALYTICAL Make a conjecture about what happens to S n as n for S n = a The third term can also be written as a 3 = and the fourth term can be written as a 4 = Upon substituting, a 4 = If we let 4 = n and 3 = n 1, the formula becomes a n = b Therefore, a recursive formula that can be used to describe the amount of medicine in a patient's system immediately following the nth dose is a 1 = d, a n = a n MULTIPLE REPRESENTATIONS In this problem, you will investigate the limits of a GRAPHICAL Graph S n = for r = 02, 05, and 09 on the same graph b TABULAR Copy and complete the table shown below c From the graph for r = 02, it appears that as n, S n 125 From the graph for r = 05, it appears that as n, S n 2 From the graph for r = 09, it appears that as n, S n 10 d esolutions Manual - Powered by Cognero Page 36

37 e It appears that as n, S n f Because 86 > 1, as n, S n 105 ERROR ANALYSIS Emilio believes that the sum of the infinite geometric series can be calculated Annie disagrees Is either of them correct? Explain your reasoning Find the common ratio 4 16 = = 025 Sample answer: The common ratio is less than 1, so the sequence is converging to 0, and the sum of the series can be calculated Therefore, Emilio is correct 106 CHALLENGE A ball is dropped from a height of 5 meters On each bounce, the ball rises to 65% of the height it reached on the previous bounce a Approximate the total vertical distance the ball travels, until it stops bouncing b The ball makes its first complete bounce in 2 seconds, that is, from the moment it first touches the ground until it next touches the ground Each complete bounce that follows takes 08 times as long as the preceding bounce Estimate the total amount of time that the ball bounces a The ball is dropped from a height of 5 meters, bounces back up 065(5) or 325 meters, falls 325 meters, bounces back up 065(325) or meters, falls meters, and so on So, an infinite sequence that can be used to represent this situation is 5, 325, 325, 21125, 21125, The corresponding series can be written as the sum of the two infinite geometric series: one series that represents the distance the ball travels when falling and one series that represents the distance the ball travels when bouncing back up Series 1 Series Find the sum of each series Therefore, the total vertical distance the ball travels is or about 236 meters b In this sequence, a 1 = 2 and r = 08 Find the sum of the related series Therefore, the total amount of time that the ball bounces is 10 seconds 107 WRITING IN MATH Explain why an infinite geometric series will not have a sum if < r 1 If then increases without limit Therefore, the corresponding sequence will be divergent, and the sum of the series cannot be calculated Consider the following geometric series is not the correct value for the series, Thus, the formula for the sum of an infinite geometric series does not work when esolutions Manual - Powered by Cognero Page 37

38 REASONING Determine whether each statement is true or false Explain your reasoning 108 If the first two terms of a geometric sequence are positive, then the third term is positive If the first two terms are positive, then the common ratio must be a positive Therefore, the third term, which equals the second term times the common ratio, must also positive So, the statement is true Consider the sequence 2, 4, The common ration is 4 2 = 2 Find the next term 4 2 = 8 Which is positive 109 If you know r and the sum of a finite geometric series, you can find the last term If the sum of the series and the common ratio are given, you can use the formula for the nth partial sum of a geometric series,, to find the last term However, the value of the first term must also be known Therefore, this statement is false 110 If r is negative, then the geometric sequence converges Whether a geometric sequence converges or diverges depends on the absolute value of the common ratio If < 1, the sequence will converge, and if > 1 the sequence will diverge Therefore, this statement is false The series 27, 9, 3, 1, converges as well as 27, 9, 3, 1, The sequence 1, 3, 9, 27, and 1, 3, 9, 27, both diverge 111 REASONING Determine whether the following statement is sometimes, always, or never true Explain your reasoning If all of the terms of an infinite geometric series are negative, then the series has a sum that is a negative number If r < 1, then the series has a sum, and the sum is a negative number If r > 1, the corresponding sequence is divergent, and the series has no sum Therefore, the statement is sometimes true Consider the series 27 + ( 9) + ( 3) + ( 1) + Consider a series with S n = 58,590 and r = 5 Find a n You can not solve this because there are two unknown a 1 and a n The sum is 405 Consider the series 1+ ( 3) + ( 9) + ( 27) + The sum is 15, which is not correct since the sum of the first 4 terms is 40 Thus, there is not sum 112 CHALLENGE The midpoints of the sides of a esolutions Manual - Powered by Cognero Page 38

39 square are connected so that a new square is formed Suppose this process is repeated indefinitely The common ratio r is < 1 Therefore, the infinite geometric series of perimeters has a sum Substitute a 1 = 16 and r = into a What is the perimeter of the square with side lengths of x inches? b What is the sum of the perimeters of all the squares? c What is the sum of the areas of all the squares? the formula for the sum of an infinite geometric series a The triangle formed by two midpoints and an included corner of the largest square is a right triangle with side lengths of 4 2 or 2 inches, and a hypotenuse of x inches Use the Pythagorean Theorem to find x Therefore, the sum of the perimeters of all the squares is or about 546 inches c The area of a square is given by A = s 2, so the area of the largest square is 4 2 or 16, and the area of the second largest square is (2 ) 2 or 8 So, a 1 The perimeter of a square is given by P = 4s, where s is the length of one side So, the perimeter of the square with side lengths of x or 2 inches is 4(2 ) or 8 inches b The perimeter of the largest square is 4(4) or 16 inches, and the perimeter of the second largest square is 8 inches So, the first two terms in the sequence of perimeters are a 1 = 16 and a 2 = 8 Find the common ratio = 16, a 2 = 8, and r = 8 16 or The common ratio r is < 1 Therefore, the infinite geometric series of areas has a sum Substitute a 1 = 16 and r = into the formula for the sum of an infinite geometric series a 2 a 1 = 8 16 or esolutions Manual - Powered by Cognero Page 39

40 for the sum of an infinite geometric series 114 Therefore, the sum of the areas of all the squares is 32 in 2 The first term of this series is 13 and the last term is 25 The number of terms is equal to the upper bound minus the lower bound plus one, which is or 5 Therefore, a 1 = 13, a n = 25, and n = 5 Find the sum of the series Find each sum The first term of this series is 3 and the last term is 15 The number of terms is equal to the upper bound minus the lower bound plus one, which is or 7 Therefore, a 1 = 3, a n = 15, and n = 7 Find the sum of the series The first term of this series is 13 and the last term is 311 The number of terms is equal to the upper bound minus the lower bound plus one, which is or 150 Therefore, a 1 = 13, a n = 311, and n = 150 Find the sum of the series esolutions Manual - Powered by Cognero Page 40

41 116 TOURIST ATTRACTIONS To prove that objects of different weights fall at the same rate, Marlene dropped two objects with different weights from the Leaning Tower of Pisa in Italy The objects hit the ground at the same time When an object is dropped from a tall building, it falls about 16 feet in the first second, 48 feet in the second second, and 80 feet in the third second, regardless of its weight If this pattern continues, how many feet would an object fall in the sixth second? In this sequence, a 1 = 16, a 2 = 48, and a 3 = 80 Find the common difference = = 32 Find a = = = 176 Therefore, the object will fall 176 feet in the sixth second 117 TEXTILES Patterns in fabric can often be created by modifying a mathematical graph The pattern can be modeled by a lemniscate a Suppose the designer wanted to begin with a lemniscate that was 6 units from end to end What polar equation could have been used? b What polar equation could have been used to generate a lemniscate that was 8 units from end to end? a The equation for a lemniscate is given by r 2 = a 2 cos 2 or r 2 = a 2 sin 2 If the lemniscate is 6 units from end to end, a = 6 2 or 3 Therefore, the corresponding equations are r 2 = 3 2 cos 2 or r 2 = 9 cos 2 and r 2 = 3 2 sin 2 or r 2 = 9 sin 2 b If the lemniscate is 8 units from end to end, a = 8 2 or 4 Therefore, the corresponding equations are r 2 = 4 2 cos 2 or r 2 = 16 cos 2 and r 2 = 4 2 sin 2 or r 2 = 16 sin 2 esolutions Manual - Powered by Cognero Page 41

42 Graph each polar equation on a polar grid 118 = The solutions of θ = are ordered pairs of the 120 = 150 The solutions of θ = 150 are ordered pairs of the form (r, 150 ), where r is any real number The graph consists of all points on the line that make an angle of 150 with the positive polar axis form, where r is any real number The graph consists of all points on the line that make an angle of with the positive polar axis 119 r = 15 The solutions of r = 15 are ordered pairs of the form (15, θ), where θ is any real number The graph consists of all points that are 15 units from the pole, so the graph is a circle centered at the origin with radius 15 esolutions Manual - Powered by Cognero Page 42

43 Find the cross product of u and v Then show that u v is orthogonal to both u and v 121 u =, v = Find the cross product of u and v 122 u =, v = Find the cross product of u and v To show that is orthogonal to both u and v, find the dot product of with u and with v To show that is orthogonal to both u and v, find the dot product of with u and with v Because both dot products are zero, the vectors are orthogonal Because both dot products are zero, the vectors are orthogonal esolutions Manual - Powered by Cognero Page 43

44 123 u =, v = Find the cross product of u and v Find the component form and magnitude of with the given initial and terminal points Then find a unit vector in the direction of 124 A(6, 7, 9), B(18, 21, 18) Find the component form of Use the component form to find the magnitude of To show that is orthogonal to both u and v, find the dot product of with u and with v Using this magnitude and component form, find a unit vector u in the direction of Because both dot products are zero, the vectors are orthogonal esolutions Manual - Powered by Cognero Page 44

45 125 A(24, 6, 16), B(8, 12, 4) Find the component form of 126 A(3, 5, 9), B( 1, 15, 7) A(3, 5, 9), B( 1, 15, 7) Find the component form of Use the component form to find the magnitude of Use the component form to find the magnitude of Using this magnitude and component form, find a unit vector u in the direction of Using this magnitude and component form, find a unit vector u in the direction of esolutions Manual - Powered by Cognero Page 45

46 127 SAT/ACT In the geometric sequence,,,, each term after the first is equal to the previous term times a constant What is the value of the 13 th term? A 2 7 B 2 8 C 2 9 D 2 10 E 2 11 = 2 = 2 Use the formula for the nth term of a geometric sequence to find a REVIEW The pattern of dots shown below continues infinitely, with more dots being added at each step Which expression can be used to determine the number of dots at the nth step? F 2n G n(n + 2) H n(n + 1) J 2(n + 1) In this sequence, a 1 = 4, a 2 = 6, and a 3 = 8 First, find the common difference 6 4 = = 2 Substitute a 1 = 4 and d = 2 in the formula for the nth term of an arithmetic sequence Therefore, the correct answer is C Therefore, the correct answer is J esolutions Manual - Powered by Cognero Page 46

47 129 The first term of a geometric series is 1, and the common ratio is 3 How many terms are in the series if its sum is 182? A 6 B 7 C 8 D 9 The first term in the series is 1, the common ratio is 3, and the sum is 182 Use the formula for the sum of a finite geometric series to find the number of terms n 130 REVIEW Cora begins a phone tree to notify her friends about a party In step 1, she calls 3 friends In step 2, each of those friends calls 3 new friends In step 3, each of those new friends calls 3 more new friends After step 3, how many people know about the party, including Cora? F 12 G 13 H 39 J = = 3 The common ratio is 3 Find the sum of the series Including Cora, 40 people know about the party Therefore, the correct answer is J Therefore, the correct answer is A esolutions Manual - Powered by Cognero Page 47