Verify/prove whether a given sequence is arithmetic or not?

A sequence is a set of numbers that follows some specific order. An arithmetic sequence is a set of numbers in which each successive number is obtained by adding a constant quantity to the previous number. Thus, in order to verify whether a sequence is arithmetic or not, you need to verify whether the difference between any two consecutive terms is constant for all terms of the sequence or not.

There are two methods to do so. These are discussed below:

Method 1

If the first three or more numbers of the sequence are given, then proceed directly, otherwise first obtain the first three numbers of the sequence. Subtract the first term from the second term and note down the result. Similarly subtract the second term from the third term and note down the result. If the two results are same, the given sequence is an arithmetic progression.

Example 1: Given sequence: 1, 3, 5, 7, ...,

Difference of first two terms = 3 - 1 = 2

Difference of second and third term = 5 - 3 = 2

Since the difference between first and second term is same as that between the second and third terms, therefore the given sequence of numbers is an arithmetic progression/sequence.

Example 2: Given sequence: T_n = n + 1

First term, T₁ = 1 + 1 = 2

Second term, T₂ = 2 + 1 = 3

Third term, T₃ = 3 + 1 = 4

Difference between first and second terms = 3 - 2 = 1

Difference between second and third terms = 4 - 3 = 1

Since the difference between first and second terms is same as that between second and third terms, therefore the given sequence is arithmetic in nature.

Method 2

This method is useful when the formula for the nth term of the sequence is given. In this method you do not obtain the first, second or third terms. Instead, you find the difference between the (n+1)th term and the nth term from the given formula. If the result is a constant number, then the given sequence is arithmetic, otherwise it is not arithmetic.

For example, given sequence T_n = 2n + 1

n^th term, T_n = n + 1

(n + 1)^th term, T_{n + 1} = (n + 1) + 1 = n + 2

Difference between (n + 1)^th term and n^th term 

= (n + 2) - (n + 1) = 1

Since 1 is a constant number, therefore the given sequence is an arithmetic progression/sequence.