If the sum of n terms of an A.P. is 2n2 + 5n, then its nth term is
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).

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