Question

The sum of first n terms of an A.P. is 5n − n². Find the n^th term of this A.P.

Answer 1

Let a be the first term and d be the common difference.

We know that, sum of first n terms = S_n = $\frac{n}{2}$[2a + (n − 1)d]

It is given that sum of the first n terms of an A.P. is 5n − n².

∴ First term = a = S₁ = 5(1) − (1)² = 4.

Sum of first two terms = S₂ = 5(2) − (2)² = 6.

∴ Second term = S₂ − S₁ = 6 − 4 = 2.

∴ Common difference = d = Second term − First term
= 2 − 4 = −2

Also, n^th term = a_n = a + (n − 1)d
⇒ a_n = 4 + (n − 1)(−2)
⇒ a_n = 4 − 2n + 2
⇒ a_n = 6 − 2n

Thus, n^th term of this A.P. is 6 − 2n.