Question

In an A.P if sum of its first $n$ terms is $3n^2+5n$ and its $k^{th}$ term is $164$, find the value of $k$.

Answer 1

Given,

$S_n=3n^2+5n$

So,

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

$=3+5$

$a_1=8$

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

$=12+10$

$=22$

So,

$a_1+a_2=22$

$8+a_2=22$

$a_2=14$

Now,

$\Rightarrow$$a=8$ and $d=14-8=6$

As, $k^{th}=164$

$\Rightarrow$$a+(k-1)d=164$

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

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

$\Rightarrow$$k-1=\frac{156}{6}$

$\Rightarrow$$k-1=26$

$\Rightarrow$$k=26+1$

$k=27$