I was asked to prove that the time complexity of merge sort is $ O(log_2n)$ but I cannot find a way to continue my method. Any help?
$T(n)=2T(\frac{n}{2} )+n$
$T(n)= 2[2T(\frac{n}{4})+n] +n = 4T(\frac{n}{4})+3n$
$T(n)=8T(\frac{n}{8})+7n$
$...$
$...$
$...$
$...$
$T(n)= 2^kT(\frac{n}{2^k})+(2^k-1)n$
Finally $\frac{n}{2^k}=1$ and $\therefore n=2^k$
I do not know how to continue from here to prove that it is $O(log_2n)$