prove by using PMI that 4 raise to n + 15n - 1 is divisible by 9 .

Let P(*n*) : 4* ^{n}* + 15

*n*– 1 is divisible by 9

**STEP I** P(1): 4^{1} + 15 × 1 – 1 = 18, which is divisible by 9

P(1) is true

**STEP II **Let P(*k*) be true. Then, 4^{k} + 15*k* – 1 is divisible by 9

⇒ 4^{k} + 15*k* – 1 = 9λ, for some λ ∈ N

We shall now show that P(*k* + 1) is true, for this we have to show that 4^{k + 1}+ 15 (*k *+ 1) – 1 is divisible by 9.

Now,

∴ P(*k* + 1) is true.

Thus, P(*k*) is true ⇒ P(*k* + 1) is true

Hence, by the principle of mathematical induction P(*n*) is true for all *n* ∈ **N** i.e., 4* ^{n}* + 15

*n*– 1 is divisible by 9.

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