I need to prove that the following recurrence relation is $O(5^n)$: $$ T(n)=5^n+3T(\lfloor n^\frac{2}{5}\rfloor) $$ And $T(n)=\Theta(1)$ for $n\le 9$.
I am trying induction, and proving that there exists $c$ such that $T(n)\le c\cdot5^n$, but I am stuck at the induction step:
$T(n)=5^n+3T(\big\lfloor n^\frac{2}{5}\big\rfloor)\le5^n+3c\cdot 5^{\lfloor n^\frac{2}{5}\rfloor}\le5^n+3c \cdot 5^{n^{\frac{2}{5}}}=\bigg(1+\frac{3c}{5^{n- n^\frac{2}{5}}}\bigg)\cdot 5^n$.
How can I continue in the induction (turn the parenthesis into the constant $c$)?