Question

Justify whether it is true to say that the sequence having following n^th term is an A.P.
(i) a_n = 2n − 1 (ii) a_n = 3n² + 5 (iii) a_n = 1 + n + n²

Answer 1

(i)

Consider the expression a_n = 2n − 1,
For n = 1, a₁ = 2(1) − 1 = 1
For n = 2, a₂ = 2(2) − 1 = 3
For n = 3, a₃ = 2(3) − 1 = 5
For n = 4, a₄ = 2(4) − 1 = 7
The first four terms are 1, 3, 5, 7.
The difference between each consecutive term is 2.
Hence this is an A.P.

(ii)
Consider the expression a_n = 3n² + 1,
For n = 1, a₁ = 3(1²) + 1 = 8
For n = 2, a₂ = 3(2²) + 1 = 17
For n = 3, a₃ = 3(3²) + 1 = 32
For n = 4, a₄ = 3(4²) + 1 = 53
The first four terms are 8, 17, 32, 53.
The difference between each consecutive is not same.
Hence this is not an A.P.

(iii)
Consider the expression a_n = 1 + n + n²,
For n = 1, a₁ = 1 + 1 + 1 = 3
For n = 2, a₂ = 1 + 2 + 4 = 7
For n = 3, a₃ = 1 + 3 + 9 = 13
For n = 4, a₄ = 1 + 4 + 16 = 21
The first four terms are 3, 7, 13, 21.
The difference between each consecutive term is not same.
Hence this is not an A.P.