Question

What term in (7^2)+2 progression isn't divisible by 3?

Answer 1

7 is congruent to 1 mod 3. Therefore all powers of 7 are congruent to 1 mod 3, meaning that all numbers of the form 7^n +2 are dibisible by 3.

Answer 2

You could also do a proof by induction:

Basecase, n=0, 7^0 + 2 = 3, clearly divisible by 3.

Assume 7^n + 2 is divisible by 3, to show that 7^(n+1) + 2 is too.

7^(n+1) + 2 = 7⋅7^n + 2 = 6⋅7^n + 7^n + 2.

As we already assumed, 7^n + 2 is divisible by 3, and 6⋅7^n is clearly divisible by 3 as well. Thus 7^n + 2 is divisible by three for all natural numbers n.

Answer 3

You mean 7^n +2, right?