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Question

Use the Principle of Mathematical Induction, check if, for any positive integer n, 6n1 is divisible by 5. True or False?


A

False

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B

True

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Solution

The correct option is B

True


For any n1, let Pn be the statement that 6n1 is divisible by 5.

Base Case––––––––––. The statement P1 says that

61 − 1 = 6 − 1 = 5

is divisible by 5, which is true.

Inductive Step–––––––––––––––. Fix k1, and suppose that Pk holds, that is, 6k1 is divisible by 5.

It remains to show that Pk+1 holds, that is, that 6k+11 is divisible by 5.

6k+11=6(6k)1

= 6(6k1)1+6

= 6(6k1)+5.

By Pk, the first term 6(6k1) is divisible by 5, the second term is clearly divisible by 5. Therefore the left hand side is also divisible by 5. Therefore Pk+1 holds.

Thus by the principle of mathematical induction, for all n1, Pn holds.


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