Question

Prove that the sum of three consecutive exponents of the number 2 is divisible by 14.

Answer 1

`2^n+2^(n+1)+2^(n+2)=2^n(1+2+2^2)=2^n\cdot7=2^(n-1)\cdot2\cdot7=2^(n-1)\cdot14`

Answer 2

So you want to prove
`14\mid2^n+2^(n+1)+2^(n+2)`. Notice that for
`n=1`, we have

`2^1+2^2+2^3=2+4+8=14`

If
`n>1`, we have

`2^n+2^(n+1)+2^(n+2)=2^(n-1)(2^1+2^2+2^3)`

and we know
`14\mid2^1+2^2+2^3`, so
`2^n+2^(n+1)+2^(n+2)` will always be a multiple of 14 and we're done.