SOLUTION: If the sum of the terms of an arithmetic series is 234, and the middle term is 26, find the number of terms in the series.

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Question 656236: If the sum of the terms of an arithmetic series is 234, and the middle term is 26, find the number of terms in the series.
Answer by jim_thompson5910(35256) (Show Source):
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The middle term is 26, so the first term is a1 = 26 - k*d for some common difference d and some number of terms before the middle term k.

The middle term is 26, so the nth term (or the last term) is an = 26 + k*d for some common difference d and some number of terms before the middle term k.

Now use the arithmetic sequence partial sum formula.

Sn = n*(a1 + an)/2

Sn = n*((26 - k*d) + (26 + k*d))/2

Sn = n*(26 - k*d + 26 + k*d)/2

Sn = n*(52)/2

234 = n*(52)/2

234*2 = 52n

468 = 52n

52n = 468

n = 468/52

n = 9

So there are 9 terms in the sequence.

Unfortunately, no unique sequence has 9 terms where they sum up to 234 (with 26 in the middle), but here are a few sequences that fit this description:

2 + 8 + 14 + 20 + 26 + 32 + 38 + 44 + 50 = 234

6 + 11 + 16 + 21 + 26 + 31 + 36 + 41 + 46 = 234

10 + 14 + 18 + 22 + 26 + 30 + 34 + 38 + 42 = 234

30 + 29 + 28 + 27 + 26 + 25 + 24 + 23 + 22 = 234

74 + 62 + 50 + 38 + 26 + 14 + 2 - 10 - 22 = 234

There are infinitely more sequences that fit this description.