If $S_n$ denotes the sum of the first $n$ terms of an AP, and $S_{10}=S_{15}$, which term must be $0$?
The $10$th term
The $12$th term
The $13$th term
The $25$th term
Step-by-Step Solution
Key Concept: When $S_p=S_q$, the terms from $(q+1)$ to $p$ (or vice versa) sum to zero, and their arithmetic mean position is zero — the standard shortcut says $a_{\frac{p+q}{2}+1}$ relates to zero sum, but more directly: terms $a_{11}$ through $a_{15}$ sum to zero, and by symmetry $a_{13}$ (their middle term) must itself be zero.
$S_{15}-S_{10}=a_{11}+a_{12}+a_{13}+a_{14}+a_{15}=0$ (since $S_{10}=S_{15}$). [0.5 Mark]
These $5$ consecutive terms are symmetric about $a_{13}$ (the middle one), so their sum equals $5a_{13}$; since this is $0$, $a_{13}=0$. [0.5 Mark]
Correct Answer: The $13$th term