Consider the following sequence: $a_j = 4a_{j-1} - 4a_{j-2}$
$a_0=0;\ a_1 = 1$

For all $j\geqslant2$, come up with a general formula for the term $a_j$. 
  Use mathematical induction to prove your claim.

I have calculated the first few terms of this sequence, and determined from them that $a_j = j · 2^{j-1}$.
However, I am very bad at induction (some might even say weak at it) and would appreciate if someone could help me prove that this is in fact the case.

For
$$A_j=4(A_{i-1}+A_{j-2}),$$
let $A_j=x^j$ to get $x^2-4x+4=0 \implies x=2,2$ (repeated roots).
So the general solution of $A_j$ is given as
$$A_j=(aj+b)x^j=(aj+b)2^j$$
Using the given initial values we get $a=1/2,b=0$
Finally $A_j=j2^{j-1}$
