Prove that 2^(80)+2^(n+1)+2^n is divisible by 7 for n belongs to the natural number.

We will prove that 2^(n+2)+2^(n+1)+2^n is divisible by 7 using formula to multiply powers with the same base:a^(b) * a^(c) = a^(b+c)Now looking at our expression we can write:2^(n+2) + 2^(n+1) + 2^n = 2^n * 2^2 + 2^n * 2^1 + 2^n * 1 = 2^n * ( 2^2 +2^1+1 ) = 2^n*(4+2+1) = 7 * 2^nTherefore 7*2^n is always divisible by 7 for n belongs to the natural numbers, because the 2^n will always be a natural number and any natural number which is multiplied by 7 will be divisible by 7.