Question

Justify whether it is true to say that the sequence having following n^th term is an A.P.

a_n = 1 + n + n²

Answer 1

Given: a_n = 1 + n + n²

Step-wise calculation:

1. Compute a_{n + 1} by replacing n with (n + 1): 

a_{n + 1} = 1 + (n + 1) + (n + 1)² 

= 1 + n + 1 + (n² + 2n + 1)

= n² + 3n + 3

2. Find the difference between consecutive terms:

a_{n + 1} – a_n = (n² + 3n + 3) – (n² + n + 1) 

= 2n + 2

3. Since a_{n + 1} – a_n = 2n + 2 depends on n (is not a constant), the difference between successive terms is not the same for all n (e.g. a₁ = 3, a₂ = 7, a₃ = 13 so differences 4 and 6 are different).

The sequence a_n = 1 + n + n² is not an arithmetic progression, because the successive difference a_{n + 1} – a_n = 2n + 2 is not constant.