# Recurrence relation complexity [duplicate]

I just learned about recurrences and I just can't solve this problem. I have this recurrence relation:

$$T(n) = \begin{cases} k\cdot T(\frac{n}{k}) & n > 0\\ 1 & n = 0\\ \end{cases}$$

where $k$ is a constant number.

I tried drawing a recurrence tree or replacing for lower $n$s but no success. I hope you can help me with an idea!

## marked as duplicate by David Richerby, D.W.♦, Juho, Rick Decker, Yuval FilmusJun 25 '15 at 1:31

• Do you really mean $T(nk)$? Is $k < 1$? – Yuval Filmus Jun 24 '15 at 18:02
Suppose that $n$ is a power of $k$, say $n = k^t$. Then $$T(k^t) = kT(k^{t-1}) = k^2T(k^{t-2}) = \cdots = k^tT(1) = k^t,$$ assuming a base case of $T(1) = 1$. So for powers of $k$, we have $T(n) = n$. You can also prove that by induction: if $T(n/k) = n/k$ then $T(n) = kT(n/k) = k(n/k) = n$.