Let there be an A.P. with first term ‘$a$’, common difference '$d$'. If $a_n$ denotes its $n^{th}$ term and $S_n$ the sum of first $n$ terms, find.$k$, if $S_n = 3n^2 + 5n$ and $a_k = 164$.

Given:

In an A.P., first term $=a$ and common difference $=d$.

$a_n$ denotes its $n^{th}$ term and $S_n$ the sum of first $n$ terms.

To do:

We have to find $k$, if $S_n = 3n^2 + 5n$ and $a_k = 164$.

Solution:

Let us substitute $n=1, 2$ to find the values of $a$ and $d$

$S_1=3(1)^2+5(1)$

$=3+5$

$=8$

$\Rightarrow a_1=a=8$

$S_2=3(2)^2+5(2)$

$=12+10$

$=22$

Second term $a_2=S_2-S_1$

$=22-8$

$=14$

Therefore,

$d=a_2-a_1$

$=14-8$

$=6$

We know that,

$n$th term $a_n=a+(n-1)d$

$a_k=a+(k-1)d$

$164=8+(k-1)6$

$164-8=(k-1)6$

$156=(k-1)6$

$k-1=26$

$k=26+1$

$k=27$

Therefore, $k=27$.