Proof by induction the divisiblity Proof by induction, that 
$$x_n=10^{(3n+2)} + 4(-1)^n\text{ is divisible by 52, when n}\in N  $$
for now I did it like that:
$$\text{for } n=0:$$
$$10^2+4=104$$
$$104/2=52$$

$$\text{Let's assume that:}$$
$$x_n=10^{(3n+2)} + 4(-1)^n=52k$$
$$\text {so else}$$
$$4(-1)^n=52k-10^{3n+2}$$

$$for \text{ n+1}:$$
$$\text {after transforms get something like that:}$$
$$52k=10^{3n+3}$$

But I'm sure, that the last step I did wrong. Actually I don't know when the proof is done, if you would help me I would be thankful.

For $n+1$ you have:
$$10^{(3n+2)+3}+4(-1)^{n+1}=10^3\cdot10^{3n+2}+4(-1)^n\cdot(-1)=\\
=10^3\cdot[52k-4(-1)^n]-4(-1)^n=10^3\cdot52k-(-1)^n(4004)$$
but $4004=52\cdot 77$, then
$$10^{(3n+2)+3}+4(-1)^{n+1}=52[10^3k-77(-1)^n]$$
