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.

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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(13342) About Me  (Show Source):
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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.