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

Let P(n) : 4ⁿ + 15n – 1 is divisible by 9

STEP I    P(1): 4¹ + 15 × 1 – 1 = 18, which is divisible by 9

P(1) is true

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

⇒ 4^k + 15k – 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., 4ⁿ + 15n – 1 is divisible by 9.