Question
Let there be an A.P. with first term ' $a$ ', common difference ' $d$ '. If $a_n$ denotes in $n^{\text {th }}$ term and $S_n$ the sum of first $n$ terms, find.k, if $S_n=3 n^2+5 n$ and $a k=164$.
✓
Answer
$a_k=S_k-S_{k-1}$
$\Rightarrow 164=\left(3 k^2+5 k\right)-\left(3(k-1)^2+5(k-1)\right)$
$\Rightarrow 164=3 k^2+5 k-3 k^2+6 k-3-5 k+5$
$\Rightarrow 164=6 k+2$
$\Rightarrow 6 k=162$
$\Rightarrow k=27$
$\Rightarrow 164=\left(3 k^2+5 k\right)-\left(3(k-1)^2+5(k-1)\right)$
$\Rightarrow 164=3 k^2+5 k-3 k^2+6 k-3-5 k+5$
$\Rightarrow 164=6 k+2$
$\Rightarrow 6 k=162$
$\Rightarrow k=27$
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