Find the sum of the first 25 terms of an A.P. whose nth term is given by an = 2 − 3n.

Question:

Find the sum of the first 25 terms of an A.P. whose nth term is given by an = 2 − 3n.

Solution:

Here, we are given an A.P. whose $n^{\text {th }}$ term is given by the following expression, $a_{n}=2-3 n$. 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.

$a=2-3(1)$

$=2-3$

 

$=-1$

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

$l=a_{n}=2-3 n$

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

$S_{25}=\left(\frac{25}{2}\right)[(-1)+2-3(25)]$

$=\left(\frac{25}{2}\right)[1-75]$

$=\left(\frac{25}{2}\right)(-74)$

$=(25)(-37)$

 

$=-925$

Therefore, the sum of the 25 terms of the given A.P. is $S_{25}=-925$.

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