If the sum to n terms of a sequence be n2 + 2n, then prove that the s

1.	If the sum to n terms of a sequence be n2 + 2n, then prove that the sequence is an A.P.
Answer» Given: S_n = n² + 2n …(i) S_n-1 = (n – 1)² + 2(n – 1) = n² + 1 – 2n + 2n – 2 = n² - 1 …(ii) Subtracting eq (ii) from (i), we get t_n = S_n – S_n-1 = n² + 2n – n² + 1 = 2n + 1 The nth term of an AP is 2n + 1.

If the sum to n terms of a sequence be n2 + 2n, then prove that the sequence is an A.P.

Answer»

Given: S_n = n² + 2n …(i)

S_n-1 = (n – 1)² + 2(n – 1) = n² + 1 – 2n + 2n – 2 = n² - 1 …(ii)

Subtracting eq (ii) from (i), we get

t_n = S_n – S_n-1 = n² + 2n – n² + 1 = 2n + 1

The nth term of an AP is 2n + 1.