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If the sum of the first n terms of an A.P. is 12(3n2 +7n), then find its nth term. Hence write its 20th term. - Mathematics

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Sum

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

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Solution

We have,

Sn = `1/2`[3n2 + 7n]

Replacing n by (n - 1), we get

Sn - 1 = `1/2`[3(n - 1)2 + 7(n - 1)]

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

⇒ Sn - 1 = `1/2`[3n2 + 3 - 6n + 7n - 7]

⇒ Sn - 1 = `1/2`[3n2 + n - 4]

Now, nth term = Sn - Sn - 1

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

⇒ an = `1/2`[3n2 + 7n - 3n2 - n + 4]

⇒ an = `1/2[6n + 4]`

⇒ an = 3n + 2

Now,

a20 = 3 × 20 + 2 = 60 + 2 = 62.

Concept: Sum of First n Terms of an AP
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