Question

The sum of the first n terms of an AP is given by S_n = (3n² – 4n). Find its

1. nth term,
2. first term and
3. common difference.

Answer 1

Given: The sum of the first n terms is S_n = (3n² – 4n).

Step-wise calculation:

i. nth term a_n:

Use a_n = S_n – S_n–1

S_n = 3n² – 4n

S_n–1 = 3(n – 1)² – 4(n – 1)

= 3n² – 10n + 7

a_n = (3n² – 4n) – (3n² – 10n + 7)

= 6n – 7

ii. First term a₁:

a₁ = a_n at n = 1

⇒ a₁ = 6(1) – 7

⇒ a₁ = –1

Check: S₁ = 3(1)² – 4(1) 

= –1, which equals a₁

iii. Common difference d:

d = a_n – a_n–1 or difference between consecutive terms of a_n = 6n – 7.

d = (6(n + 1) – 7) – (6n – 7)

d = 6