Question

If the sum of first n terms of an A.P. is given by `S_n = n/2 (2n + 8)`. Then, find its first term and common difference. Hence, find its 15^th term.

Answer 1

Given the sum of n terms of an A.Р. is `S_n = n/2 (2n + 8)`.

To find the first term and common difference by putting the value of n = 1, 2.

Putting the value of n = 1

`S_1 = 1/2 (2 xx 1 + 8)`

⇒ `S_1 = (2 + 8)/2`

= `10/2`

= 5

We know that S₁ = a₁

So, the first term of A.P. = a₁ = 5

Putting value of n = 2

`S_2 = 2/2 [2 xx 2 + 8]`

S₂ = 1[4 + 8]

S₂ = 12

We know that,

S₂ = a₁ + a₂

⇒ 12 = 5 + a₂

⇒ a₂ = 12 – 5

⇒ a₂ = 7

Common difference, d = a₂ – a₁

= 7 – 5

= 2

15^th term of A.P. = a_n  = a + (n – 1) × d

⇒  a₁₅ = 5 + (15 – 1) × 2

= 5 + 14 × 2

= 5 + 28

= 33

Therefore, the first term of the A.P. is 5, the common difference is 2 and 15^th term is 33.