Question

The sum of the first n terms of an arithmetic sequence is 2n² + 3n. find the algebraic form of this sequence.

Answer 1

Sum of first n terms of an arithmetic sequence = 2n² + 3n

First term of the arithmetic sequence = x₁ = 2(1)² + 3(1)

= 2 + 3

= 5

Sum of first two terms = x₁ + x₂ = 2(2)² + 3(2)

= 8 + 6

= 14

⇒ 5 + x₂ = 14

⇒ x₂ = 14 − 5 = 9

Sum of first three terms = x₁ + x₂ + x₃ = 2(3)² + 3(3)

= 18 + 9

= 27

⇒ 5 + 9 + x₃ = 27

⇒ x₃ = 27 − 14 = 13

Thus, the arithmetic sequence is 5, 9, 13, …

Here, a + b = 5 and common difference = a = 9 − 5 = 4

⇒ 4 + b = 5

⇒ b = 5 − 4 = 1

Therefore,

Thus, the algebraic form of the given sequence is 4n + 1, where n is a natural number.