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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):
You can put this solution on YOUR website! 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.
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