# SOLUTION: In an arithmetic series the sum of the first 9 terms is 162 and the sum of the first 12 terms is 288. Determine the series.

Algebra ->  Algebra  -> Sequences-and-series -> SOLUTION: In an arithmetic series the sum of the first 9 terms is 162 and the sum of the first 12 terms is 288. Determine the series.      Log On

 Ad: Algebrator™ solves your algebra problems and provides step-by-step explanations! Ad: Algebra Solved!™: algebra software solves algebra homework problems with step-by-step help!

 Algebra: Sequences of numbers, series and how to sum them Solvers Lessons Answers archive Quiz In Depth

 Question 257916: In an arithmetic series the sum of the first 9 terms is 162 and the sum of the first 12 terms is 288. Determine the series.Answer by Theo(3464)   (Show Source): You can put this solution on YOUR website!sum of the terms of an arithmetic series is given by the formula: s[n] = n * (a[1] + a[n])/2 s[n] = sum of the n terms of the sequence. a[1] = first term in the sequence. a[n] = nth term in the sequence. n = number of terms in the sequence. the formuula for the nth term in the sequence is: a[n] = a[1] + (n-1) * d d is the common difference. you will use both these formulas to find the answer. you are given: sum of the first 9 terms in a sequence is 162 sum of the first 12 terms in the same sequence is 288. the formnula for the sum of the first n terms in a sequence is: s[n] = n * (a[1] + a[n])/2 substitute for the first 9 terms to get: 162 = 9 * (a[1] + a[9])/2 (first equation) substitute for the first 12 terms to get: 288 = 12 * (a[1] + a[12])/2 (second equation) multiply both sides of the first equation by 2 and divide both sides of the first equation by 9 to get: a[1] + a[9] = 36 (third equation) multiply both sides of the second equation by 2 and divide both sides of the second equation by 12 to get: a[1] + a[12] = 48 (fourth equation) subtract third equation from fourth equation to get: a[12] - a[9] = 12 since 12 - 9 = 3, there are 3 terms between a[9] and a[12]. 12/3 = 4 which means the common difference between each term is 4. the nth term in each sequence is given by the formula: a[n] = a[1] + (n-1) * d we now know that d = 4 we solve for a[9] to get: a[9] = a[1] + 8*4 = a[1] + 32 we take our first equation and plug in our new found values. first equation is: 162 = 9 * (a[1] + a[9])/2 (first equation) substitute a[1] + 32 for a[9] to get: 162 = 9 * (a[1] + a[1] + 32)/2 combine like terms to get: 162 = 9 * (2* a[1] + 32)/2 multiply both sides of this equation by 2 and divide both sides of this equation by 9 to get: 36 = 2 * a[1] + 32 subtract 32 from both sides of this equation to get: 36-32 = 2 * a[1] which becomes: 4 = 2 * a[1]. divide both sides of this equation by 2 to get: a[1] = 2. we now know a[1] and we know d so we can construct the sequence. formula for nth term in the sequence is: a[n] = a[1] + (n-1) * d since d = 4,... the 9th term in the sequence is: a[9] = 2 + 8*4 = 2 + 32 = 34 the 12th term in the sequence is: a[12] = 2 + 11*4 = 2 + 44 = 46 the sequence is: ``` term: 1 2 3 4 5 6 7 8 9 10 11 12 sequence: 2 6 10 14 18 22 26 30 34 38 42 46 ``` 2 is the 1st term. 34 is the 9th term. 46 is the 12th term. common difference is 4.