Question

In an AP, if $S_n=3n^2+5n$ and $a_k=164$, find the value of k.

Answer 1

$S_n=3n^2+5n$
$\therefore S_1=a=3+5=8$
and $S_2=a_1+a_2=3\times 2^2+5\times 2=22$
$\Rightarrow a+a+d=22\Rightarrow 2a+d=22$
$\Rightarrow d=22-2a=22-16=6$
Now $a_k=164$
$\Rightarrow a+(k-1)d=164$
$\Rightarrow 8+(k-1)6=164$
$\Rightarrow (k-1)6=156$
$\Rightarrow k-1=26\Rightarrow k=27$