The number $4^{n} + 15 n - 1$ is a multiple of $9$ for any natural $n$ .

True

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False

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Solution

The correct option is A True
$P (n) : 4^{n} + 15 n - 1$ is a multiple of 9 for any natural number n.
$4^{n}$ can be written as $(1 + 3)^{n}$ whose binomial expansion gives :
$1 + 3 n + 9 (^{n} C_{2}) + . . . . . . . . + 3^{n} + 15 n - 1$
which reduces to :
$18 n + 9 (^{n} C_{2}) + . . . . . . . . + 3^{n} = 9 k$
where k is an integer.
Therefore, $4^{n} + 15 n - 1$ is divisible by 9.

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