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

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

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Given: Sn = n2 + 2n …(i)

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

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

tn = Sn – Sn-1 = n2 + 2n – n2 + 1 = 2n + 1

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