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 $92$, what is the third term in the GP?

• A. $33$
• B. $46$
• C. $92$
• D. $108$

Answer 1

Answer: (C)

$92$

Sum of first $12$ terms is equal to sum of first $14$ terms.
Sum of first $14$ terms $=$ Sum of first $12$ terms $+$ ${13}^{th}$ term $+{14}^{th}$ term
$\therefore {13}^{th}term+{14}^{th}term=0$.
Let us assume ${13}^{th}$ term $=$ k, common ratio $=r$. ${14}^{th}$ term will be $kr$.
$k+kr=0$
$k(1+r)=0$
$\therefore r=-1$ 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 $19$ terms is $92$
$\therefore a=92$
Third term $=a=92$