Question:
If the sum of $n$ terms of an A.P. is $2 n^{2}+5 n$, then its $n$th term is
(a) 4n − 3
(b) 3n − 4
(c) 4n + 3
(d) 3n + 4
Solution:
Here, the sum of first n terms is given by the expression,
$S_{n}=2 n^{2}+5 n$
We need to find the nth term.
So we know that the nthterm of an A.P. is given by,
So,
$a_{n}=\left(2 n^{2}+5 n\right)-\left[2(n-1)^{2}+5(n-1)\right]$
Using the property,
$(a-b)^{2}=a^{2}+b^{2}-2 a b$
We get,
$a_{v}=\left(2 n^{2}+5 n\right)-\left[2\left(n^{2}+1-2 n\right)+5(n-1)\right]$
$=\left(2 n^{2}+5 n\right)-\left(2 n^{2}+2-4 n+5 n-5\right)$
$=2 n^{2}+5 n-2 n^{2}-2+4 n-5 n+5$
$=4 n+3$
Therefore, $a_{n}=4 n+3$
Hence the correct option is (c).