Question

Show that 3 · 4 n + 51 is divisible by 3 and 9 for all positive integers n.

Answer 1

Answer:

Yes, it is

Step-by-step explanation:

3 * 4^n + 51 = 3 * (4^n + 17),

which is divisible by 3.

I will prove that 4^n + 17 is divisible by 3

Case 1: n = 1

4^1 + 17 = 21 , which is divisible by 3.

Let n = k, a positive integer, such that

4^k + 17 is divisible by 3

If n = k + 1, then

4^(k + 1) + 17 = 4 * 4^k + 17

= (4^k + 17) + 3 * 4^k

4^k + 17 and 3 * 4^k are both divisible by 3,

so 4^k + 17 + 3 * 4^k is divisible by 3 as well.

Since 4^n + 17 is divisible by 3 for any positive integer n, then

3 * (4^n + 17) would be divisible by 9.