0 votes
in Mathematical Induction by (52.3k points)
closed by

Using the principle of mathematical induction, prove each of the following for all n ϵ N: 

(23n – 1) is a multiple of 7

1 Answer

+1 vote
by (56.6k points)
selected by
Best answer

To Prove:

23n – 1 is a multiple of 7

Let us prove this question by principle of mathematical induction (PMI) for all natural numbers

23n – 1 is a multiple of 7

Let P(n): 23n – 1 which is a multiple of 7

For n = 1 P(n) is true since 23 – 1 = 8 - 1 = 7, which is multiple of 7

Assume P(k) is true for some positive integer k , ie,

= 23k – 1 = 7m, where m ∈ N …(1)

We will now prove that P(k + 1) is true whenever P( k ) is true 


[Adding and subtracting 23]

= 7 x r , where r = 2m + 1 is a natural number

Therefore 23n - 1 is multiple of 7 

Therefore, P (k + 1) is true whenever P(k) is true 

By the principle of mathematical induction, P(n) is true for all natural numbers ie, N 

Hence proved

Welcome to Sarthaks eConnect: A unique platform where students can interact with teachers/experts/students to get solutions to their queries. Students (upto class 10+2) preparing for All Government Exams, CBSE Board Exam, ICSE Board Exam, State Board Exam, JEE (Mains+Advance) and NEET can ask questions from any subject and get quick answers by subject teachers/ experts/mentors/students.