Question

Question 23 Prove the following by using the principle of mathematical induction for all n∈N: 41^n – 14^n is a multiple of 27.

Class X1 - Maths -Principle of Mathematical Induction Page 95

Answer 1

41ⁿ - 14ⁿ is a multiple of 27
First of all,
Let p(n):41ⁿ-14ⁿ is a multiple of 27.

step1:- for n = 1
P(1): 41¹-14¹ = 27
This is multiple of 27.so, It is true for n = 1 .

step2:- for n = k
p(k): 41^k - 14^k
Let 27L = 41^k - 14^k -----(1)

step3 :- for n = k+1
P(k+1):41^(k+1)-14^(k+1)
= 41^k.41 - 14^k.14
from eqns (1)
= (27L+14^k)41 - 14^k.14
= 27L×41 + 14^k(41 - 14)
= 27L×41 + 14^k.27
= 27( 41L + 14^k)
Here, it is clear that P(k+1) is multiple of 27. it means p(k+1) is true when p(k) is true . hence, from the principle of mathematical induction, the statement is true.