Question

The sum of ‘n’ terms of a progression is n(n + 1). Prove that it is an A.P. Also, find its 10th term.

Answer 1

The sum of n terms is given by S_n = n(n + 1). The n^th term can be found using the formula:

`a_n = S_n - S_(n - 1)`

a_n = [n(n + 1)] − [(n − 1) ((n − 1) + 1)]

a_n = (n² + n) − [(n − 1)(n)]

a_n = n² + n − (n² − n)

a_n = n² + n − n² + n

a_n = 2n

To prove it is an A.P., we find the common difference d by calculating `a_n − a_(n - 1)`

d = `a_n − a_(n - 1)`

d = 2n − 2(n − 1)

d = 2n − 2n + 2

d = 2

Using the general term formula, a_n = 2n

n = 10

a₁₀ = 2(10)

a₁₀ = 20