Question

Prove that no matter what the real numbers a and b are, the sequence with nth term a + nb is always an A.P. What is the common difference?

Answer 1

Hint: In order to prove that the given sequence is AP and to find the common difference, we use the general formula of the nth term of an Arithmetic Progression. Then we compare the nth term given in the question to the general form to find the first term and the common difference.

Complete step-by-step answer:
Given Data,
Real numbers a and b are in A.P.
Nth term is a + nb.

A.P means Arithmetic Progression. An Arithmetic progression is defined as the sequence of numbers such that the difference between any two consecutive numbers is always constant.
The formula of nth term of an A.P is given by:
 $ {{\text{a}}_{\text{n}}} = {{\text{a}}_1} + \left( {{\text{n - 1}}} \right){\text{d}} $ --- (1)
Where $ {{\text{a}}_{\text{n}}} $ is the nth term, $ {{\text{a}}_1} $ is the first term, n is the number of terms and d is the common difference of the A.P respectively.

Now given that the nth term of the series is,
$ {{\text{a}}_{\text{n}}} = {\text{a}} + {\text{nb}} $ --- (2)

Equation (1) can be also written as, $ {{\text{a}}_{\text{n}}} = {{\text{a}}_1} + {\text{nd - d}} $

Now comparing equations (1) and (2), we get
 $ {{\text{a}}_{\text{n}}} = {{\text{a}}_1} + {\text{nd - d}} $
 $ {{\text{a}}_{\text{n}}} = {\text{a}} + {\text{nb}} $
Common difference d = b (by comparing the coefficients of n)
First term $ {{\text{a}}_1}{\text{ - d = a}} $
Substituting d = b in the above we get,
 $ \Rightarrow {{\text{a}}_1}{\text{ = a + b}} $ (First term of AP)

Hence if a and b are real numbers the sequence forms Arithmetic Progression (A.P) with the first term of the sequence as, $ {{\text{a}}_1}{\text{ = a + b}} $ and has a common difference d = b.

Note: In order to solve this type of questions the key is to know the concept of Arithmetic Progression and the formulae of general terms in an AP like nth term of the equation, common difference, first term of an AP and sum of all terms of an AP.
It is important to expand the general form of the nth term of AP and rearrange it, to easily compare it to the given term to deduce the first term and common difference of the given series. There are other forms of progressions in mathematics, like geometric progression and Harmonic progression.