Find the sum of the first 25 terms of an A.P.

Here, we are given an A.P. whose $n^{\text {th }}$ term is given by the following expression, $a_{n}=7-3 n$. We need to find the sum of first 25 terms.

So, here we can find the sum of the $n$ terms of the given A.P., using the formula, $S_{n}=\left(\frac{n}{2}\right)(a+l)$

Where, a = the first term

l = the last term

So, for the given A.P,

The first term (a) will be calculated using in the given equation for nth term of A.P.

$a=7-3(1)$

$=7-3$

$=4$

Now, the last term (l) or the nth term is given

$l=a_{n}=7-3 n$

So, on substituting the values in the formula for the sum of n terms of an A.P., we get,

$S_{25}=\left(\frac{25}{2}\right)[(4)+7-3(25)]$

$=\left(\frac{25}{2}\right)[11-75]$

$=\left(\frac{25}{2}\right)(-64)$

$=(25)(-32)$

$=-800$

Therefore, the sum of the 25 terms of the given A.P. is $S_{n}=-800$.