Homework Problem Set Sample Solutions

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1 Homework Problem Set Sample Solutions S.73 S The recursive formula for a geometric sequence is aa nn+11 = (aa nn ) with aa 00 = Find an explicit formula for this sequence. The common ratio is , and the initial value is , so the explicit formula is aa nn = ( ) nn for nn The explicit formula for a geometric sequence is aa nn = 11(22. 11) Find a recursive formula for this sequence. First, we rewrite the sequence as aa nn = nn = 11( ) nn. We then see that the common ratio is , and the initial value is 11, so the recursive formula is aa nn+11 = ( )aa nn with aa 00 = The first term aa 00 of a geometric sequence is 55, and the common ratio rr is 22. a. What are the terms aa 00, aa 11, and aa 22? aa 00 = 55 aa 11 = aa 22 = 2222 Scaffolding: Students may need the hint that in the Opening Exercise, they wrote the terms of a geometric sequence so they can begin with the first three terms of such a sequence and use it to find rr. b. Find a recursive formula for this sequence. The recursive formula is aa nn+11 = 22aa nn, with aa 00 = 55. c. Find an explicit formula for this sequence. The explicit formula is aa nn = 55( 22) nn, for nn 00. d. What is term aa 99? Using the explicit formula, we find: aa 99 = ( 55) ( 22) 99 = e. What is term aa? One solution is to use the explicit formula: aa = ( 55) ( 22) = Another solution is to use the recursive formula: aa = aa 99 ( 22) = : Geometric Sequences and Exponential Growth and Decay Unit 3: Exponential Functions 151

2 S Term aa 44 of a geometric sequence is , and term aa 55 is a. What is the common ratio rr? We have rr = = The common ratio is b. What is term aa 00? From the definition of a geometric sequence, aa 44 = aa 00 rr 44 = , so aa 00 = ( 11.11) 44 = = c. Find a recursive formula for this sequence. The recursive formula is aa nn+11 = (aa nn ) with aa 00 = 44. d. Find an explicit formula for this sequence. The explicit formula is aa nn = 44( ) nn, for nn If a geometric sequence has aa 11 = and aa 88 = , find the exact value of the common ratio rr. The recursive formula is aa nn+11 = aa nn rr, so we have aa 88 = aa 77 (rr) = aa 66 rr 22 = aa 55 rr 33 = aa 11 rr = rr = rr rr = 22 : Geometric Sequences and Exponential Growth and Decay Unit 3: Exponential Functions 152

3 6. If a geometric sequence has aa 22 = and aa 66 = , approximate the value of the common ratio rr to four decimal places. The recursive formula is aa nn+11 = aa nn rr, so we have aa 66 = aa 55 (rr) = aa 44 rr 22 = aa 33 rr 33 = aa 22 rr = rr 44 rr 44 = rr = Find the difference between the terms aa of an arithmetic sequence and a geometric sequence, both of which begin at term aa 00 and have aa 22 = 44 and aa 44 =. Arithmetic: The explicit formula has the form aa nn = aa 00 + nnnn, so aa 22 = aa and aa 44 = aa Then aa 44 aa 22 = 44 = 88 and aa 44 aa 22 = (aa ) (aa ), so that 88 = 2222 and dd = 44. Since dd = 44, we know that aa 00 = aa = = 44. So, the explicit formula for this arithmetic sequence is aa nn = We then know that aa = = Geometric: The explicit formula has the form aa nn = aa 00 (rr nn ), so aa 22 = aa 00 (rr 22 ) and aa 44 = aa 00 rr 44, so aa 44 rr 22 and aa 22 aa 44 = = 33. Thus, aa rr22 = 33, so rr = ± 33. Since rr 22 = 33, we have aa 22 = 44 = aa 00 rr 22, so that aa 00 = 44. Then the explicit formula for this geometric sequence is aa 33 nn = ± 33 nn. We then know that aa = ± 33 = = = Thus, the difference between the terms aa of these two sequences is = aa nn+11 = aa nn with aa 00 = 11 : Geometric Sequences and Exponential Growth and Decay Unit 3: Exponential Functions 153

4 S Given the geometric sequence defined by the following values of aa 00 and rr, find the value of nn so that aa nn has the specified value. a. aa 00 = 6666, rr = 11 22, aa nn = 22 The explicit formula for this geometric sequence is aa nn = nn and aa nn = = nn 22 Thus, aa 55 = 22. nn = = 11 nn 22 nn = 55 b. aa 00 =, rr = 33, aa nn = The explicit formula for this geometric sequence is aa nn = (33) nn, and we have aa nn = Thus, aa 88 = (33) nn = nn = nn = nn = 88 c. aa 00 = , rr = , aa nn = The explicit formula for this geometric sequence is aa nn = (11. 99) nn, and we have aa nn = Thus, aa = (11. 99) nn = (11. 99) nn =. 99 nn llllll(11. 99) = llllll(. 99) lloogg(. 99) nn = = llllll(11. 99) : Geometric Sequences and Exponential Growth and Decay Unit 3: Exponential Functions 154

5 d. aa 00 = 11, rr = , aa nn = The explicit formula for this geometric sequence is aa nn = 11(00. 77) nn, and we have aa nn = Thus, aa = (00. 77) nn = (00. 77) nn = nn = 9. Jenny planted a sunflower seedling that started out 55 cccc tall, and she finds that the average daily growth is cccc. a. Find a recursive formula for the height of the sunflower plant on day nn. hh nn+11 = hh nn with hh 00 = 55 b. Find an explicit formula for the height of the sunflower plant on day nn 00. hh nn = MP Kevin modeled the height of his son (in inches) at age nn years for nn = 22, 33,, 88 by the sequence hh nn = (nn 22). Interpret the meaning of the constants 3333 and in his model. At age 22, Kevin s son was 3333 inches tall, and between the ages of 22 and 88 he grew at a rate of inches per year. : Geometric Sequences and Exponential Growth and Decay Unit 3: Exponential Functions 155

6 S.77 MP Astrid sells art prints through an online retailer. She charges a flat rate per order for an order processing fee, sales tax, and the same price for each print. The formula for the cost of buying nn prints is given by PP nn = a. Interpret the number in the context of this problem. The number represents a $ order processing fee. b. Interpret the number. 66 in the context of this problem. The number. 66 represents the cost of each print, including the sales tax. c. Find a recursive formula for the cost of buying nn prints. PP nn = PP nn 11 with PP 11 =. (Notice that it makes no sense to start the sequence with nn = 00, since that would mean you need to pay the processing fee when you do not place an order.) 12. A bouncy ball rebounds to 9999% of the height of the preceding bounce. Craig drops a bouncy ball from a height of 2222 feet. a. Write out the sequence of the heights hh 11, hh 22, hh 33, and hh 44 of the first four bounces, counting the initial height as hh 00 = hh 11 = hh 22 =. 22 hh 33 = hh 44 =. 11 b. Write a recursive formula for the rebound height of a bouncy ball dropped from an initial height of 2222 feet. hh nn+11 = hh nn with hh 00 = 2222 c. Write an explicit formula for the rebound height of a bouncy ball dropped from an initial height of 2222 feet. hh nn = 2222(00. 99) nn for nn 00 : Geometric Sequences and Exponential Growth and Decay Unit 3: Exponential Functions 156

7 d. How many bounces does it take until the rebound height is under 66 feet? 2222(00. 99) nn < 66 (00. 99) nn = nn nn > So, it takes bounces for the bouncy ball to rebound under 66 feet. e. Extension: Find a formula for the minimum number of bounces needed for the rebound height to be under yy, feet, for a real number 00 < yy < S (00. 99) nn < yy 13. Show that when a quantity aa 00 = AA is increased by %, its new value is aa 11 = AA If this quantity is again increased by %, what is its new value aa 11 22? If the operation is performed nn times in succession, what is the final value of the quantity aa nn? We know that % of a number AA is represented by AA. Thus, when aa = AA is increased by %, the new quantity is aa 11 = AA + 11 AA = AA If we increase it again by %, we have aa 22 = aa aa 11 = aa 11 = aa 00 = aa 00. If we repeat this operation nn times, we find that aa nn = 11 + nn 11 aa 00. : Geometric Sequences and Exponential Growth and Decay Unit 3: Exponential Functions 157

8 14. When Eli and Daisy arrive at their cabin in the woods in the middle of winter, the interior temperature is 4444 FF. a. Eli wants to turn up the thermostat by 22 FF every minutes. Find an explicit formula for the sequence that represents the thermostat settings using Eli s plan. Let nn represent the number of -minute increments. The function EE(nn) = models the thermostat settings using Eli s plan. b. Daisy wants to turn up the thermostat by 44% every minutes. Find an explicit formula for the sequence that represents the thermostat settings using Daisy s plan. Let nn represent the number of -minute increments. The function DD(nn) = 4444( ) nn models the thermostat settings using Daisy s plan c. Which plan gets the thermostat to 6666 F most quickly? Making a table of values, we see that Eli s plan sets the thermostat to 6666 FF first. nn Elapsed Time EE(nn) DD(nn) minutes minutes minutes minutes hour hour minutes hour 3333 minutes hour 4444 minutes hours hours minutes hours 3333 minutes : Geometric Sequences and Exponential Growth and Decay Unit 3: Exponential Functions 158

9 d. Which plan gets the thermostat to 7777 FF most quickly? Continuing the table of values from part (c), we see that Daisy s plan sets the thermostat to 7777 FF first. nn Elapsed Time EE(nn) DD(nn) 22 hours 4444 minutes hours hours minutes hours 3333 minutes hours 4444 minutes : Geometric Sequences and Exponential Growth and Decay Unit 3: Exponential Functions 159