In an AP, if $ S_{n}=3 n^{2}+5 n $ and $ a_{k}=164 $, find the value of $ k $.
Given:
In an AP, \( S_{n}=3 n^{2}+5 n \) and \( a_{k}=164 \)
To do:
We have to find $k$.
Solution:
Let $a$ be the first term and $d$ be the common difference.
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, $k=27$.
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