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, 4k + 15k – 1 is divisible by 9

⇒ 4k + 15k – 1 = 9λ, for some λ ∈ N

We shall now show that P(k + 1) is true, for this we have to show that 4k + 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 nN i.e., 4n + 15n – 1 is divisible by 9.

  • 56
What are you looking for?