Using the principle of mathematical induction prove that `41^n-14^n` is a multiple of `27` .

Explanation

Let P (k) br true for some positive integer k, i.e.,

`41^k−14^k` is a multiple of 27
`∴ 41^k−14^k=27m, . . . . .(1)`

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

Consider

`41^(k+1)−14^(k+1)`

`=41^k.41−14^k.14`

`=41(41^k−14^k+14^k)−14^k.14`

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