So you want to prove [tex]14\mid2^n+2^{n+1}+2^{n+2}[/tex]. Notice that for [tex]n=1[/tex], we have
[tex]2^1+2^2+2^3=2+4+8=14[/tex]
If [tex]n>1[/tex], we have
[tex]2^n+2^{n+1}+2^{n+2}=2^{n-1}(2^1+2^2+2^3)[/tex]
and we know [tex]14\mid2^1+2^2+2^3[/tex], so [tex]2^n+2^{n+1}+2^{n+2}[/tex] will always be a multiple of 14 and we're done.
