The sum of first m terms of an AP is 4m² – m. If its n^th term is 107, find the value of n. Also find the 21th term of this A.P.

Sum of first m terms = 4m² – m

Putting m = 1, we get

S₁ = 4(1)² – 1 = 3

Putting m = 2, we get

Sum of first two terms = S₂ = 4(2)² – 2 = 14

⸫ The second term = 14 – 3 = 11

Putting m = 3, we get

S₃ = 4(3)² – 3 = 33

⸫ The third term = 33 – 14 = 19

⸫ First term = a = 3

⸫ Common difference = d = 8

The formula to find the n^th term of an A.P. is given by,

a_n = a + (n – 1) d

⇒ a_n = a + (n – 1) d = 107

⇒ 3 + (n – 1) 8 = 107

⇒ 3 + 8n – 8 = 107

⇒ 8n = 112

⸫ n = 14

⸫ a₂₁ = 3 + (21 – 1) 8 = 163