using principle of mathematical induction prove that 4^n-3n-1 is a multiple of 9 - Maths - Principle of Mathematical Induction

Question

using principle of mathematical induction prove that 4^n-3n-1 is a
multiple of 9

Neha Sethi · Accepted Answer

4n-3n-1Let P(n) be the given statementNow,P(n):4n-3n-1 is a multiple of 9Step 1: P(1): 41-3(1)-1=0It is a multiple of 9Step 2: Let P(m) be trueThen 4m-3m-1 is a multple of 9Suppose 4m-3m-1=3k where k∈NWe have to show that P(m+1) is true whenever P(m) is true
Now,P(m+1)=4m+1-3m+1-1=4m 4-3m-3-1=4m 4-3m-4=22m
22-3m-4=422m-3m4-1=422m-3m-1+9m4Since the hypothesis is true for m,22m-3m-1=9k for some integer k
So, 49k+9m4=36k+9m=9(4k+1) which is a multiple of 9Theerfore, the hypothesis is true for m+1
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