Question

Prove that 8^n+8^(n+1)+8^(n+2) is divisible by 584, n is natural number. How can I do this without using mathematical induction?

Answer 1

Write \(\large 8^{n+1}\) as \(\large 8^n\cdot 8\) (do you see why you can do that, using the rules of exponents?).

Do similarily with \(\large 8^{n+2}\).

You'll then be able to factor the whole expression in a way that will be very helpful. :)

Answer 2

Also, you can write \(\large 8^n=8\cdot 8^{n-1}\).

You'll need that in the factored form as well.