Question

The nth term of a progression is (n² − n + 1). Prove that it is not an A.P.

Answer 1

Let Tn = n² − n + 1. For an A.P., the difference Tn + 1 − Tn

Tn + 1 − Tn = [(n+1)² − (n+1) + 1] − [n² − n + 1]

= (n² + 2n + 1 − n − 1 + 1) − (n² − n + 1)

= (n² + n + 1) − (n² − n + 1)

= 2n.

Since 2n depends on n, the consecutive-term difference is not constant. Therefore, the sequence is not an arithmetic progression.