# Find the sum to n terms of the series whose n^{th} term is given by n^{2} + 2^{n}

**Solution:**

The given n^{th} term is a_{n} = n^{2} + 2^{n}

Hence,

S_{n} = ∑^{n}_{k = 1}(k^{2} + 2k)

= ∑^{n}_{k = 1}(k)^{2} + ∑^{n}_{k = 1}(2k) ....(1)

Consider ∑^{n}_{k = 1}(2k) = 2^{1} + 2^{2} + 2^{3} + ....

The above series 2^{1} + 2^{2} + 2^{3} + .... is a G.P with both the first term and common ratio equal to 2.

∑^{n}_{k = 1}(2k) = (2) [(2)^{n} - 1]/(2 - 1)

= 2 (2^{n} - 1) ....(2)

From (1) and (2) , we obtain

S_{n} = ∑^{n}_{k = 1}(k^{2} + 2 (2^{n} - 1))

= n/6 (n + 1)(2n + 1) + 2 (2^{n} - 1)

NCERT Solutions Class 11 Maths Chapter 9 Exercise 9.4 Question 9

## Find the sum to n terms of the series whose n^{th} term is given by n^{2} + 2^{n}

**Solution:**

Therefore, the sum to n terms of the series whose nth term is given by n^{2} + 2^{n} is n/6 (n + 1)(2n + 1) + 2 (2^{n} - 1)