11.4 Partial Sums of Arithmetic and Geometric Sequences
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1 Section.4 Partial Sums of Arithmetic and Geometric Sequences 653 Integrated Review SEQUENCES AND SERIES Write the first five terms of each sequence, whose general term is given. 7. a n = n a n = + n 3. a n = 3 n - 4. a n = n 2-5 Find the indicated term for each sequence n ; a n 2 + 2; a n ; a n n 2n ; a 4 Write the first five terms of the arithmetic or geometric sequence, whose first term is a and whose common difference, d, or common ratio, r, are given. 9. a = 7; d = a = -3; r = 5. a = 45; r = 3 2. a = -2; d = 0 Find the indicated term of each sequence. 3. The tenth term of the arithmetic sequence whose first term is 20 and whose common difference is 9 4. The sixth term of the geometric sequence whose first 5. The seventh term of the geometric sequence 6, -2, 24, c 6. The twentieth term of the arithmetic sequence -00, -85, -70, c 7. The fifth term of the arithmetic sequence whose fourth term is -5 and whose tenth term is The fifth term of a geometric sequence whose fourth term is and whose seventh term is 8 Evaluate a 5i 20. a 3i + 22 i= 2. a 7 i=3 i= 2 i a 5 i=2 i i + Find each partial sum. 23. Find the sum of the first three terms of the sequence whose general term is a n = nn Find the sum of the first ten terms of the sequence whose general term is a n = -2 n n + 2. term is 64 and whose common ratio is Partial Sums of Arithmetic and Geometric Sequences S Find the Partial Sum of an Arithmetic Sequence. 2 Find the Partial Sum of a Geometric Sequence. 3 Find the Sum of the Terms of an Infinite Geometric Sequence. Finding Partial Sums of Arithmetic Sequences Partial sums S n are relatively easy to find when n is small that is, when the number of terms to add is small. But when n is large, finding S n can be tedious. For a large n, S n is still relatively easy to find if the addends are terms of an arithmetic sequence or a geometric sequence. For an arithmetic sequence, a n = a + n - 2d for some first term a and some common difference d. So S n, the sum of the first n terms, is S n = a + a + d2 + a + 2d2 + g + a + n - 2d2 We might also find S n by working backward from the nth term a n, finding the preceding term a n -, by subtracting d each time. S n = a n + a n - d2 + a n - 2d2 + g + a n - n - 2d2 Now add the left sides of these two equations and add the right sides. 2S n = a + a n 2 + a + a n 2 + a + a n 2 + g+ a + a n 2 The d terms subtract out, leaving n sums of the first term, a, and last term, a n. Thus, we write 2S n = na + a n 2 or S n = n 2 a + a n 2
2 654 CHAPTER Sequences, Series, and the Binomial Theorem Partial Sum S n of an Arithmetic Sequence The partial sum S n of the first n terms of an arithmetic sequence is given by S n = n 2 a + a n 2 where a is the first term of the sequence and a n is the nth term. EXAMPLE Use the partial sum formula to find the sum of the first six terms of the arithmetic sequence 2, 5, 8,, 4, 7, c. Solution Use the formula for S n of an arithmetic sequence, replacing n with 6, a with 2, and a n with 7. S n = n 2 a + a n 2 S 6 = = 392 = 57 2 Use the partial sum formula to find the sum of the first five terms of the arithmetic sequence 2, 9, 6, 23, 30,.... EXAMPLE 2 Find the sum of the first 30 positive integers. Solution Because, 2, 3, c, 30 is an arithmetic sequence, use the formula for S n with n = 30, a =, and a n = 30. Thus, S n = n 2 a + a n 2 S 30 = = 532 = Find the sum of the first 50 positive integers. EXAMPLE 3 Stacking Rolls of Carpet Rolls of carpet are stacked in 20 rows with 3 rolls in the top row, 4 rolls in the next row, and so on, forming an arithmetic sequence. Find the total number of carpet rolls if there are 22 rolls in the bottom row. 3 rolls 4 rolls 5 rolls Solution The list 3, 4, 5, c, 22 is the first 20 terms of an arithmetic sequence. Use the formula for S n with a = 3, a n = 22, and n = 20 terms. Thus, S 20 = = 0252 = 250 There are a total of 250 rolls of carpet. 3 An ice sculptor is creating a gigantic castle-facade ice sculpture for First Night festivities in Boston. To get the volume of ice necessary, large blocks of ice were stacked atop each other in 0 rows. The topmost row comprised 6 blocks of ice, the next row 7 blocks of ice, and so on, forming an arithmetic sequence. Find the total number of ice blocks needed if there were 5 blocks in the bottom row.
3 Section.4 Partial Sums of Arithmetic and Geometric Sequences Finding Partial Sums of Geometric Sequences We can also derive a formula for the partial sum S n of the first n terms of a geometric series. If a n = a r n -, then S n = a + a r + a r 2 + g + a r n - c c c c st 2nd 3rd nth term term term term Multiply each side of the equation by -r. -rs n = -a r - a r 2 - a r 3 - g - a r n Add the two equations. S n - rs n = a + a r - a r2 + a r 2 - a r a r 3 - a r g - a r n S n - rs n = a - a r n Now factor each side. S n - r2 = a - r n 2 Solve for S n by dividing both sides by - r. Thus, S n = a - r n 2 - r as long as r is not. Partial Sum S n of a Geometric Sequence The partial sum S n of the first n terms of a geometric sequence is given by S n = a - r n 2 - r where a is the first term of the sequence, r is the common ratio, and r. EXAMPLE 4 Find the sum of the first six terms of the geometric sequence 5, 0, 20, 40, 80, 60. Solution Use the formula for the partial sum S n of the terms of a geometric sequence. Here, n = 6, the first term a = 5, and the common ratio r = 2. S n = a - r n 2 - r S 6 = = = 35 4 Find the sum of the first five terms of the geometric sequence 32, 8, 2, 2, 8. EXAMPLE 5 Finding Amount of Donation A grant from an alumnus to a university specified that the university was to receive $800,000 during the first year and 75% of the preceding year s donation during each of the following 5 years. Find the total amount donated during the 6 years. Solution The donations are modeled by the first six terms of a geometric sequence. Evaluate S n when n = 6, a = 800,000, and r = S 6 = 800,000[ ] = +2,630, The total amount donated during the 6 years is $2,630,
4 656 CHAPTER Sequences, Series, and the Binomial Theorem 5 A new youth center is being established in a downtown urban area. A philanthropic charity has agreed to help it get off the ground. The charity has pledged to donate $250,000 in the first year, with 80% of the preceding year s donation for each of the following 6 years. Find the total amount donated during the 7 years. 3 Finding Sums of Terms of Infinite Geometric Sequences Is it possible to find the sum of all the terms of an infinite sequence? Examine the partial sums of the geometric sequence 2, 4, 8, c. S = 2 S 2 = = 3 4 S 3 = = 7 8 S 4 = = 5 6 S 5 = = 3 32 f S 0 = g = Even though each partial sum is larger than the preceding partial sum, we see that each partial sum is closer to than the preceding partial sum. If n gets larger and larger, then S n gets closer and closer to. We say that is the limit of S n and that is the sum of the terms of this infinite sequence. In general, if r 6, the following formula gives the sum of the terms of an infinite geometric sequence. Sum of the Terms of an Infinite Geometric Sequence The sum S of the terms of an infinite geometric sequence is given by S = a - r where a is the first term of the sequence, r is the common ratio, and r 6. If r Ú, S does not exist. What happens for other values of r? For example, in the following geometric sequence, r = 3. 6, 8, 54, 62, c Here, as n increases, the sum S n increases also. This time, though, S n does not get closer and closer to a fixed number but instead increases without bound.
5 Section.4 Partial Sums of Arithmetic and Geometric Sequences 657 EXAMPLE 6 Find the sum of the terms of the geometric sequence 2, 2 3, 2 9, 2 27, c. Solution For this geometric sequence, r =. Since r 6, we may use the formula 3 for S of a geometric sequence with a = 2 and r = 3. S = a - r = 2-3 = = 3 6 Find the sum of the terms of the geometric sequence 7, 7 4, 7 6, 7 64, c. The formula for the sum of the terms of an infinite geometric sequence can be used to write a repeating decimal as a fraction. For example, = g This sum is the sum of the terms of an infinite geometric sequence whose first term a is 3 0 and whose common ratio r is 0. Using the formula for S, S = a - r = = 3 So, = 3. EXAMPLE 7 Distance Traveled by a Pendulum On its first pass, a pendulum swings through an arc whose length is 24 inches. On each pass thereafter, the arc length is 75% of the arc length on the preceding pass. Find the total distance the pendulum travels before it comes to rest. Solution We must find the sum of the terms of an infinite geometric sequence whose first term, a, is 24 and whose common ratio, r, is Since r 6, we may use the formula for S. S = a - r = = = 96 The pendulum travels a total distance of 96 inches before it comes to rest. 7 The manufacturers of the perpetual bouncing ball claim that the ball rises to 96% of its dropped height on each bounce of the ball. Find the total distance the ball travels before it comes to rest if it is dropped from a height of 36 inches.
6 658 CHAPTER Sequences, Series, and the Binomial Theorem Vocabulary, Readiness & Video Check Decide whether each sequence is geometric or arithmetic.. 5, 0, 5, 20, 25, ; 2. 5, 0, 20, 40, 80, ; 3., 3, 9, 27, 8 ; 4.,, 3, 5, 7, ; 5. 7, 0, 7, 4, 2, ; 6. 7, 7, 7, 7, 7, ; Martin-Gay Interactive Videos See Video.4 Watch the section lecture video and answer the following questions From Example, suppose you are asked to find the sum of the first 00 terms of an arithmetic sequence in which you were given only the first few terms. You need the 00th term for the partial sum formula how can you find this term without actually writing down the first 00 terms? 8. From the lecture before Example 2, we know r in the partial sum formula because it would make the denominator 0. What would a geometric sequence with r = look like? How could the partial sum of such a sequence be found (without the given formula)? 9. From the lecture before Example 3, why can t you find S for the geometric sequence -, -3, -9, -27, c?.4 Exercise Set Use the partial sum formula to find the partial sum of the given arithmetic or geometric sequence. See Examples and 4.. Find the sum of the first six terms of the arithmetic sequence, 3, 5, 7, c. 2. Find the sum of the first seven terms of the arithmetic sequence -7, -, -5, c. 3. Find the sum of the first five terms of the geometric sequence 4, 2, 36, c. 4. Find the sum of the first eight terms of the geometric sequence -, 2, -4, c. 5. Find the sum of the first six terms of the arithmetic sequence 3, 6, 9, c. 6. Find the sum of the first four terms of the arithmetic sequence -4, -8, -2, c. 7. Find the sum of the first four terms of the geometric sequence 2, 2 5, 2 25, c. 8. Find the sum of the first five terms of the geometric sequence 3, - 2 3, 4 3, c. Solve. See Example Find the sum of the first ten positive integers. 0. Find the sum of the first eight negative integers.. Find the sum of the first four positive odd integers. 2. Find the sum of the first five negative odd integers. Find the sum of the terms of each infinite geometric sequence. See Example , 6, 3, c 4. 45, 5, 5, c 5. 0, 00, 000, c , 3 20, 3 80, c 7. -0, -5, - 5, c , -4, -, c 9. 2, - 4, 32, c , 3 5, , c , - 3, 6, c 22. 6, -4, 8 3, c MIXED Solve. 23. Find the sum of the first ten terms of the sequence -4,, 6, c, 4 where 4 is the tenth term. 24. Find the sum of the first twelve terms of the sequence -3, -3, -23, c, -3 where -3 is the twelfth term. 25. Find the sum of the first seven terms of the sequence 3, 3 2, 3 4, c. 26. Find the sum of the first five terms of the sequence -2, -6, -8, c. 27. Find the sum of the first five terms of the sequence -2, 6, -3, c.
7 Section.4 Partial Sums of Arithmetic and Geometric Sequences Find the sum of the first four terms of the sequence - 4, - 3 4, - 9 4, c. 29. Find the sum of the first twenty terms of the sequence 2, 7 7, 0, c, - where - is the twentieth term Find the sum of the first fifteen terms of the sequence -5, -9, -3, c, -6 where -6 is the fifteenth term. 3. If a is 8 and r is - 2 3, find S If a is 0, a 8 is 3 2, and d is - 2, find S 8. Solve. See Example Modern Car Company has come out with a new car model. Market analysts predict that 4000 cars will be sold in the first month and that sales will drop by 50 cars per month after that during the first year. Write out the first five terms of the sequence and find the number of sold cars predicted for the twelfth month. Find the total predicted number of sold cars for the first year. 34. A company that sends faxes charges $3 for the first page sent and $0.0 less than the preceding page for each additional page sent. The cost per page forms an arithmetic sequence. Write the first five terms of this sequence and use a partial sum to find the cost of sending a nine-page document. 35. Sal has two job offers: Firm A starts at $22,000 per year and guarantees raises of $000 per year, whereas Firm B starts at $20,000 and guarantees raises of $200 per year. Over a 0-year period, determine the more profitable offer. 36. The game of pool uses 5 balls numbered to 5. In the variety called rotation, a player who sinks a ball receives as many points as the number on the ball. Use an arithmetic series to find the score of a player who sinks all 5 balls. Solve. See Example A woman made $30,000 during the first year she owned her business and made an additional 0% over the previous year in each subsequent year. Find how much she made during her fourth year of business. Find her total earnings during the first four years. 38. In free fall, a parachutist falls 6 feet during the first second, 48 feet during the second second, 80 feet during the third second, and so on. Find how far she falls during the eighth second. Find the total distance she falls during the first 8 seconds. 39. A trainee in a computer company takes 0.9 times as long to assemble each computer as he took to assemble the preceding computer. If it took him 30 minutes to assemble the first computer, find how long it takes him to assemble the fifth computer. Find the total time he takes to assemble the first five computers (round to the nearest minute). 40. On a gambling trip to Reno, Carol doubled her bet each time she lost. If her first losing bet was $5 and she lost six consecutive bets, find how much she lost on the sixth bet. Find the total amount lost on these six bets. Solve. See Example A ball is dropped from a height of 20 feet and repeatedly rebounds to a height that is 4 of its previous height. Find the 5 total distance the ball covers before it comes to rest. 20 ft 42. A rotating flywheel coming to rest makes 300 revolutions in the first minute and in each minute thereafter makes 2 5 as many revolutions as in the preceding minute. Find how many revolutions the wheel makes before it comes to rest. MIXED Solve. 43. In the pool game of rotation, player A sinks balls numbered to 9, and player B sinks the rest of the balls. Use an arithmetic series to find each player s score (see Exercise 36). 44. A godfather deposited $250 in a savings account on the day his godchild was born. On each subsequent birthday, he deposited $50 more than he deposited the previous year. Find how much money he deposited on his godchild s twentyfirst birthday. Find the total amount deposited over the 2 years. 45. During the holiday rush, a business can rent a computer system for $200 the first day, with the rental fee decreasing $5 for each additional day. Find the fee paid for 20 days during the holiday rush. 46. The spraying of a field with insecticide killed 6400 weevils the first day, 600 the second day, 400 the third day, and so on. Find the total number of weevils killed during the first 5 days. 47. A college student humorously asks his parents to charge him room and board according to this geometric sequence: $0.0 for the first day of the month, $0.02 for the second day, $0.04 for the third day, and so on. Find the total room and board he would pay for 30 days. 48. Following its television advertising campaign, a bank attracted 80 new customers the first day, 20 the second day, 60 the third day, and so on in an arithmetic sequence. Find how many new customers were attracted during the first 5 days following its television campaign.
8 660 CHAPTER Sequences, Series, and the Binomial Theorem REVIEW AND PREVIEW Evaluate. See Section # 5 # 4 # 3 # 2 # # 7 # 6 # 5 # 4 # 3 # 2 # # 2 # 2 # 5 # 4 # 3 # 2 # 3 # 2 # Multiply. See Section x x x x CONCEPT EXTENSIONS 57. Write as an infinite geometric series and use the formula for S to write it as a rational number. 58. Write as an infinite geometric series and use the formula S to write it as a rational number. 59. Explain whether the sequence 5, 5, 5, c is arithmetic, geometric, neither, or both. 60. Describe a situation in everyday life that can be modeled by an infinite geometric series..5 The Binomial Theorem S Use Pascal s Triangle to Expand Binomials. 2 Evaluate Factorials. 3 Use the Binomial Theorem to Expand Binomials. 4 Find the nth Term in the Expansion of a Binomial Raised to a Positive Power. In this section, we learn how to expand binomials of the form a + b2 n easily. Expanding a binomial such as a + b2 n means to write the factored form as a sum. First, we review the patterns in the expansions of a + b2 n. a + b2 0 = term a + b2 = a + b 2 terms a + b2 2 = a 2 + 2ab + b 2 3 terms a + b2 3 = a 3 + 3a 2 b + 3ab 2 + b 3 4 terms a + b2 4 = a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + b 4 5 terms a + b2 5 = a 5 + 5a 4 b + 0a 3 b 2 + 0a 2 b 3 + 5ab 4 + b 5 6 terms Notice the following patterns.. The expansion of a + b2 n contains n + terms. For example, for a + b2 3, n = 3, and the expansion contains 3 + terms, or 4 terms. 2. The first term of the expansion of a + b2 n is a n, and the last term is b n. 3. The powers of a decrease by for each term, whereas the powers of b increase by for each term. 4. For each term of the expansion of a + b2 n, the sum of the exponents of a and b is n. (For example, the sum of the exponents of 5a 4 b is 4 +, or 5, and the sum of the exponents of 0a 3 b 2 is 3 + 2, or 5.) Using Pascal s Triangle There are patterns in the coefficients of the terms as well. Written in a triangular array, the coefficients are called Pascal s triangle. a + b2 0 : a + b2 : a + b2 2 : a + b2 3 : a + b2 4 : a + b2 5 : n = 0 n = n = 2 n = 3 n = 4 n = 5
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