Show that $9|2^{2n+1}+2^{4n+1}-4$ 
Show that $9|2^{2n+1}+2^{4n+1}-4$ ( n is a positive integer):
  1-Using induction
  2-Don't use induction  

I have noticed that $2^m \equiv 2^{m+6} \pmod 9 $ but couldn't use it to solve the problem!

$2x^2+2x-4=2(x^2+x-2)=2(x+2)(x-1)=2(x-1+3)(x-1)$ so, for $x=2^{2n}=4^n$,
$$
2^{2n+1}+2^{4n+1}-4=2(4^n-1+3)(4^n-1)
$$
and you just need to show that $3\mid 4^n-1$. 
No induction
Since $4\equiv1\pmod{3}$, the statement is proved.
With induction
The base case is obvious. If $4^n-1=3k$ with $k$ integer,
$$
4^{n+1}-1=4\cdot 4^n-1=4(3k+1)=12k+4-1=3(4k+1)
$$
