Question

The value of the sum 7 x 11 + 11 x 15 + 15 x 19 + ..... + 95 x 99 is

Answer 1

The required sum of given series = 80707

Step-by-step explanation:

• In such type of problems, we try to find out the $n^t^h$ term of the series.

• And, then apply sigma operator on it. We know that sigma operator $\Sigma$ is known as summation operator and pi operator $\Pi$ is known as multiplication operator.

• In this way; we write the $n^t^h$ term as;

• $n^t^h$ term of the given series = $(4n+3)\times (4n+7)$

       = $16n^2 + 40n + 21$

• $\Sigma (n^t^h \ term) = 16 \Sigma n^2 +40\Sigma n + 21\Sigma 1$

       = $\dfrac{16n(n+1)(2n+1)}{6} + \dfrac{40n(n+1)}{2} +21n$

• To find out the value of n,

     We pick the Arithmetic progression 7, 11, 15, ........., 95.

     We find that there are total 23 terms in this A.P.

     Therefore; n = 23

Now, put n = 23 in above formula, we get.

• $\Sigma t_n = \dfrac{16\times 23\times 24\times 47}{6} + (20\times 23\times 24) + (21\times 23)$

• $\Sigma t_n = 80707$

Thus, the required sum of given series = 80707