I've been given a problem with the following sequence:
$$a_n = \begin{cases}0, & n=0 \\ a_{\lfloor n/3\rfloor} + a_{\lfloor 2n/5\rfloor} + X \cdot n & {n \geq 1}\end{cases} $$
where $X \in \mathbb{N}$ is an unknown constant. I'm supposed to use strong constructive induction to find the largest possible $X$ such that $(\forall n \in \mathbb{N})[a_n \leq 20 \cdot n] $
So far I have the following:
Basis:
Let $n=0$. Then by the definition of $a$, $a_0 = 0$, so $P(0) \Leftrightarrow 0 \leq 20 \cdot 0 \Leftrightarrow 0 \leq 0$.
Inductive Hypothesis:
For $n \geq 1$, assume $P(i)$, $\forall i \in \{0 \ldots n-1 \} \Leftrightarrow a_i \leq 20 \cdot i $
Inductive Step
This is where I'm not sure how to proceed. I am used to problems that gave me something like $a_n \leq C \cdot n$ and then I had to find $C$, which in this case would have been $20$, but I am not sure how to go about finding $X$.
Any advice or hints are greatly appreciated. Thank you!