10-2 Arithmetic Sequences and Series. Write an equation for the nth term of each arithmetic sequence. 29. SOLUTION: to find the nth term.
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1 29 Write an equation for the nth term of each arithmetic sequence 32 CCSS STRUCTURE José averaged 123 total pins per game in his bowing league this season He is taking bowling lessons and hopes to bring his average up by 8 pins each new season a Write an equation to represent the nth term of the sequence b If the pattern continues, during what season will José average 187 per game? c Is it reasonable for this pattern to continue indefinitely? Explain Use the value of a 1 to find the nth term a Given d = 8 and a 1 = 123 Find the nth term b Substitute 187 for a n and solve for n Therefore José s average will be 187 pins per game in the 9 th season c Sample answer: No; there are a maximum of 300 points in a bowling game, so it would be impossible for the average to continue to climb indefinitely esolutions Manual - Powered by Cognero Page 1
2 Find the arithmetic means in each sequence 38 Here a 1 = 182 and a 7 = the first 300 even natural numbers Here a 1 = 2 and a 300 = 300 n = 300 Therefore, the missing numbers are (182 13) or 169, (169 13) or 156, (156 13) or 143, (143 13) or 130, and (130 13) or n = 19, a n = 154, d = 8 Find the sum of each arithmetic series 41 the first 100 odd natural numbers Find the value of a 1 Here a 1 = 1 and a 100 = 199 n = 100 esolutions Manual - Powered by Cognero Page 2
3 Find the first three terms of each arithmetic series 54 a 1 = 19, n = 44, S n = a 1 = 33, n = 36, S n = 6372 Find the value of a n Find the value of a n Find the value of d Find the value of d Therefore, the first three terms are 33, 21 and 9 Therefore, the first three terms are 19, 28 and 37 esolutions Manual - Powered by Cognero Page 3
4 56 PRIZES A radio station is offering a total of $8500 in prizes over ten hours Each hour, the prize will increase by $100 Find the amounts of the first and last prize 57 Find the sum of each arithmetic series Given n = 10, d = 100 and S 10 = 8500 There are or 16 terms, so n = 16 Find the value of a 1 Find the value of a 10 Therefore, esolutions Manual - Powered by Cognero Page 4
5 58 60 There are or 10 terms, so n = 10 There are or 13 terms, so n = 13 Therefore, Therefore, 59 There are or 12 terms, so n = 12 Therefore, esolutions Manual - Powered by Cognero Page 5
6 Use the given information to write an equation that represents the nth term in each arithmetic sequence 63 The 100th term of the sequence is 245 The common difference is The sixth term of the sequence is 34 The 23rd term is 119 Given a 6 = 34 and a 23 = 119 Given a 100 = 245, d = 13 and n = 100 Find the value of a 1 Therefore, there are ( ) or 18 terms between 34 and 119 Find the common difference of the series with a 1 = 34 and a 18 = 119 Substitute the values of a 1 and d to find the nth term Find the value of a 1 64 The eleventh term of the sequence is 78 The common difference is 9 Substitute the values of a 1 and d to find the nth term Given a 11 = 78, d = 9 and n = 11 Find the value of a 1 Substitute the values of a 1 and d to find the nth term esolutions Manual - Powered by Cognero Page 6
7 66 The 25th term of the sequence is 121 The 80th term is 506 Given a 25 = 121 and a 80 = CCSS MODELING The rectangular tables in a reception hall are often placed end-to-end to form one long table The diagrams below show the number of people who can sit at each of the table arrangements Therefore, there are ( ) or 56 terms between 121 and 506 Find the common difference of the series with a 1 = 121 and a 56 = 506 a Make drawings to find the next three numbers as tables are added one at a time to the arrangement b Write an equation representing the nth number in this pattern c Is it possible to have seating for exactly 100 people with such an arrangement? Explain Find the value of a 1 a For each increase in the number of table, the number of people who can sit is increased by 4 That is, the common difference is 4 Therefore, the next three numbers are (10 + 4) or 14, (14 + 4) or 18 and (18 + 4) or 22 Substitute the values of a 1 and d to find the nth term b Substitute a 1 = 6 and d = 4 in c No; there is no whole number n for which esolutions Manual - Powered by Cognero Page 7
8 70 SPORTS While training for cross country, Silvia plans to run 3 miles per day for the first week, and then increase the distance by a half mile each of the following weeks a Write an equation to represent the nth term of the sequence b If the pattern continues, during which week will she be running 10 miles per day? c Is it reasonable for this pattern to continue indefinitely? Explain 72 Find the value of x There are x or x 2 terms, so n = x 2 a Given a 1 = 3 and d = 05 Find the nth term Equate the sum with the given value and solve for x b Substitute 10 for a n in and solve for n During 15th week, she will be running 10 miles per day c Sample answer: No; eventually the number of miles per day will become unrealistic The value of x should be positive Therefore, x = 18 esolutions Manual - Powered by Cognero Page 8
9 73 There are x or x 4 terms, so n = x 4 74 CCSS CRITIQUE Eric and Juana are determining the formula for the nth term for the sequence 11, 2, 7, 16, Is either of them correct? Explain your reasoning Equate the sum with the given value and solve for x Sample answer: Eric; Juana missed the step of multiplying d by n 1 The value of x should be positive Therefore, x = 16 esolutions Manual - Powered by Cognero Page 9
10 75 REASONING If a is the third term in an arithmetic sequence, b is the fifth term, and c is the eleventh term, express c in terms of a and b Given a 3 = a, a 5 = b and a 11 = c Find the common difference Find the value of a 1 Find the value of c in terms of a and b esolutions Manual - Powered by Cognero Page 10
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