Question

The n^th term of an A.P. cannot be n² + 1. Justify your answer.

Answer 1

Given: Let the n^th term be a_n = n² + 1.

Step-wise calculation:

1. For an A.P. the difference a_{n + 1} – a_n must be constant (independent of n); equivalently the n^th term of an A.P. is a linear function of n.

2. Compute the successive difference:

a_{n + 1} – a_n = [(n + 1)² + 1] – [n² + 1] 

= (n² + 2n + 1 + 1) – (n² + 1)

= 2n + 1

3. 2n + 1 depends on n (e.g. for n = 1 it is 3, for n = 2 it is 5), so the difference is not constant.

Since the common difference is not constant, {a_n} is not an A.P.; therefore the n^th term of an A.P. cannot be n² + 1.