Question

Prove that the nth term of an AP cannot be n² + 1. Justify your answer.

Answer 1

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.
a_n is the n ^th term of an A.P. if a_n – a_{n - 1} = constant
Given, a_n = n ² + 1
a_n – a_{n -1} = (n ² + 1) – [(n – 1)² + 1]
= (n ² + 1) – (n ² – 2n + 2)
= 2n – 1
∴ a_n – a_{n -1} ≠ constant
Thus, a_n = n ² + 1 cannot be the n ^th term of A.P.