Question

The sum of first n terms of an A.P. is given by S_n = 3n² + 2n. Determine the A.P. and its 15th term.

Answer 1

S_n = 3n² + 2n

Taking n = 1, we get

S₁ = 3(1)² + 2(1)

⇒ S₁ = 3 + 2

⇒ S₁ = 5

⇒ a₁ = 5

Taking n = 2, we get

S₂ = 3(2)² + 2(2)

⇒ S₂ = 12 + 4

⇒ S₂ = 16

∴ a₂ = S₂ – S₁ = 16 – 5 = 11

Taking n = 3, we get

S₃ = 3(3)² + 2(3)

⇒ S₃ = 27 + 6

⇒ S₃ = 33

∴ a₃ = S₃ – S₂ = 33 – 16 = 17

So, a = 5,

d = a₂ – a₁ = 11 – 5 = 6

Now, we have to find the 15^th term

a_n = a + (n – 1)d

a₁₅ = 5 + (15 – 1)6

a₁₅ = 5 + 14 × 6

a₁₅ = 5 + 84

a₁₅ = 89

Hence, the 15^th term is 89 and AP is 5, 11, 17, 23,…