Question

If the sum of the first n terms of an A.P. is `1/2`(3n² +7n), then find its n^th term. Hence write its 20^th term.

Answer 1

We have,

S_n = `1/2`[3n² + 7n]

Replacing n by (n - 1), we get

S_{n - 1} = `1/2`[3(n - 1)² + 7(n - 1)]

⇒ S_{n - 1} = `1/2`[3(n² + 1 - 2n) + 7n - 7]

⇒ S_{n - 1} = `1/2`[3n² + 3 - 6n + 7n - 7]

⇒ S_{n - 1} = `1/2`[3n² + n - 4]

Now, nth term = S_n - S_{n - 1}

⇒ a_n = `1/2[3n^2 + 7n] - 1/2[3n^2 + n - 4]`

⇒ a_n = `1/2`[3n² + 7n - 3n² - n + 4]

⇒ a_n = `1/2[6n + 4]`

⇒ a_n = 3n + 2

Now,

a₂₀ = 3 × 20 + 2 = 60 + 2 = 62.