Find the sum to n term of the A.P. 5, 2, −1, −4, −7, ...,

In the given problem, we need to find the sum of the $n$ terms of the given A.P. ${ }^{*} 5,2,-1,-4,-7, \ldots$ ".

So, here we use the following formula for the sum of n terms of an A.P.,

$S_{n}=\frac{n}{2}[2 a+(n-1) d]$

Where; a = first term for the given A.P.

d = common difference of the given A.P.

n = number of terms

For the given A.P. $(5,2,-1,-4,-7, \ldots)$,

Common difference of the A.P. $(d)=a_{2}-a_{1}$

$=2-5$

$=-3$

Number of terms (n) = n

First term for the given A.P. (a) = 5

So, using the formula we get,

$S_{n}=\frac{n}{2}[2(5)+(n-1)(-3)]$

$=\frac{n}{2}[10+(-3 n+3)]$

$=\frac{n}{2}[10-3 n+3]$

$=\frac{n}{2}[13-3 n]$

Therefore, the sum of first $n$ terms for the given A.P. is $\left.\frac{n}{2}[13-3 n]\right]$.