We have $T(n) \leq T(\lceil \frac{n}{5} \rceil) + T(\lceil \frac{7n}{10} \rceil)$. Show that $T(n) < c'n$ for all $n$ and for some constant $c'$.
Looks straightforward enough, but suprisingly I get stuck at the base case:
For $n=1$ we have: $T(1)\leq 2T(1)+cn \Leftrightarrow -T(1) \leq cn $
Which doesn't help me much.
Is $T(n)$ defined for $n\in\mathbb{N}$?
Show that $\lceil\frac{n}{5}\rceil\leq\frac{n+4}{5}$ and similar for the other term.
Find an $N$ for which,
$\,\,$ If $n>N$, and $T(k)\leq c\,'k$ for all $k\leq n$, then $T(n+1)\leq c\,'(n+1)$ as well.
Lastly, find a $c\,'$ that works for all $k=1..N$ I think it will depend on the values of $T(1), T(2)$ and $T(3)$ but on only a finite number of $T(n)$