Is there a way to find the sum of an infinite series (not geometric)

It’s a general fact that

$$\sum_{k\ge 0}\binom{k+n}nx^k=\frac1{(1-x)^{n+1}}\;.$$

You can prove this by induction on $n$, starting with the geometric series

$$\frac1{1-x}=\sum_{k\ge 0}x^k$$

and differentiating repeatedly with respect to $x$. You want the case $n=3$:

$$\sum_{k\ge 0}\binom{k+3}kx^k=\sum_{k\ge 0}\binom{k+3}3x^k=\frac1{(1-x)^4}\,.$$

Now just substitute $x=0.2$.