Find the sum to n terms of the series 1 x 2 + 2 x 3 + 3 x 4 + 4 x 5 + ....

Solution:

The given series is 1 x 2 + 2 x 3 + 3 x 4 + 4 x 5 + .... n terms.

Hence,

a_n = n (n + 1)

Therefore,

S_n = ∑ⁿ_{k = 1}(a)_k = ∑ⁿ_{k = 1}k(k + 1)

∑ⁿ_{k = i}(k)² + ∑ⁿ_{k = 1}k

= [n (n +1)(2n +1)]/6 + [n (n + 1)]/2]

= n (n + 1)]/2 x [(2n +1)/3 + 1]

= (n + 1)]/2 x (2n + 4)/3

= n/3 (n + 1)(n + 2)

NCERT Solutions Class 11 Maths Chapter 9 Exercise 9.4 Question 1

Find the sum to n terms of the series 1 x 2 + 2 x 3 + 3 x 4 + 4 x 5 + ....

Summary:

Therefore, the sum to n terms of the series 1 x 2 + 2 x 3 + 3 x 4 + 4 x 5 + .... n terms is n/3 (n + 1)(n + 2)