I'm working out a recursion tree and I've got a pattern that goes:
$$ 5 \, n^2 $$ $$ 4 \, \left(\frac{5 \, n}{2} \right)^2 $$ $$ 16 \, \left( \frac{5 \, n}{4} \right)^2. $$
I can figure out the summation pattern for $5n^2$ easy enough, as the pattern is simply $(5n/2^i)^2$. What I'm having trouble with is the multiplication factor. It appears to be recursively defined as the previous factor (starting at $i=1$) multiplied by $4$.
You have $$ \begin{split} a_0 &= 5n^2\\ a_1 &= 4(5n/2)^2 = 4 \frac{5^2 n^2}{2^2} = 5^2 n^2\\ a_2 &= 16(5n/4)^2 = 4 \frac{5^2 n^2}{4^2} = 5^2 n^2 \end{split} $$ The sequence $5n^2, 5^2n^2, 5^2n^2$ is definitely not a beginning of a geometric sequence. After the first term it could be a constant sequence.