Find the sum of first 20 terms of the sequence whose nth term is an = An + B.

Question:

Find the sum of first 20 terms of the sequence whose nth term is an = An + B.

Solution:

Here, we are given an A.P. whose nth term is given by the following expression. We need to find the sum of first 20 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.

$a=A(1)+B$

$=A+B$

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

$l=a_{n}=A n+B$

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

$S_{20}=\left(\frac{20}{2}\right)[(A+B)+A(20)+B]$

$=10[21 A+2 B]$

$=210 A+20 B$

Therefore, the sum of the first 20 terms of the given A.P. is $S_{20}=210 A+20 B$.