JEE Main & Advanced Mathematics Binomial Theorem and Mathematical Induction Question Bank Self Evaluation Test - Principle of Mathematical Induction

For all \[n\in N,\,\,\,{{41}^{n}}-{{14}^{n}}\] is a multiple of

A) 26

B) 27

C) 25

D) None of these

Correct Answer: B

Solution :

[b] Let P(n) be the statement given by \[P(n):{{41}^{n}}-{{14}^{n}}\] is a multiple of 27

For \[n=1,\]

i.e., \[P(1)={{41}^{1}}-{{14}^{1}}=27=1\times 27,\]

which is a multiple of 27.

\[\therefore \,\,\,\,P(1)\] is true.

Let \[P(k)\] be true, i.e, \[{{41}^{k}}-{{14}^{k}}=27\lambda \] (i)

For \[n=k+1,\]

\[{{41}^{k+1}}-{{14}^{k+1}}={{41}^{k}}41-{{14}^{k}}14\]

\[=(27\lambda +{{14}^{k}})41-{{14}^{k}}14\] [using (i)]

\[=(27\lambda \times 41)+({{14}^{k}}\times 41)-({{14}^{k}}\times 14)\]

\[=(27\lambda \times 41)+{{14}^{k}}(41-14)=(27\lambda \times 41)+({{14}^{k}}\times 27)\]\[=27(41\lambda +{{14}^{k}}),\]

which is a multiple of 27.

Therefore, P(K+1) is true when P (K) is true hence. From the principle of mathematical induction, the statement is true for all natural numbers n.