Common difference of an A.P. must always be a constant.
∴ d cannot be n – 1. Here, d varies when n takes different values.
For n = 1, d = 1 – 1 = 0
For n = 2, d = 2 – 1 = 1
For n = 3, d = 3 – 1 = 2
∴ d is not constant.
Thus, d cannot be taken as n – 1.
an is the n th term of an A.P. if an – an - 1 = constant
Given, an = n 2 + 1
an – an -1 = (n 2 + 1) – [(n – 1)2 + 1]
= (n 2 + 1) – (n 2 – 2n + 2)
= 2n – 1
∴ an – an -1 ≠ constant
Thus, an = n 2 + 1 cannot be the n th term of A.P.