Find the sum to n terms of the series whose n^th term is given by n² + 2ⁿ

Solution:

The given n^th term is a_n = n² + 2ⁿ

Hence,

S_n = ∑ⁿ_{k = 1}(k² + 2k)

= ∑ⁿ_{k = 1}(k)² + ∑ⁿ_{k = 1}(2k) ....(1)

Consider ∑ⁿ_{k = 1}(2k) = 2¹ + 2² + 2³ + ....

The above series 2¹ + 2² + 2³ + .... 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 (2ⁿ - 1) ....(2)

From (1) and (2) , we obtain

S_n = ∑ⁿ_{k = 1}(k² + 2 (2ⁿ - 1))

= n/6 (n + 1)(2n + 1) + 2 (2ⁿ - 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 n² + 2ⁿ

Solution:

Therefore, the sum to n terms of the series whose nth term is given by n² + 2ⁿ is n/6 (n + 1)(2n + 1) + 2 (2ⁿ - 1)