Question

Sum of first 12 terms of a GP is equal to the sum of the first 14 terms in the same GP. Sum of the first 17 terms is 92, then the third term in the GP is ___

Answer 1

Sum of first 14 terms = Sum of first 12 terms + 13th term + 14th term $\Rightarrow {13}^{th}$ term + ${14}^{th}$ term = 0
Let us assume ${13}^{th}$ term = k, common ratio = r. ${14}^{th}$ term will be kr.
k + kr = 0
k (1 + r) = 0
$\Rightarrow$ r = -1 as k cannot be zero
Common ratio = -1.
Now, if the first term of this GP is a, second term would be -a, third would be a and so on
The GP would be a, -a, a, -a, a, -a,...
Sum to even number of terms = 0
Sum to odd number of terms = a
Sum to 17 terms is $92\Rightarrow$ a = 92
Third term = a = 92