# What is the solution to this recursion?

Take $a_0=10^6$.

What is $a_n$ (asymptotically) where $a_{i+1}=a_i+\sqrt[\alpha]{a_i}$ where $\alpha>1$?

How fast does $a_n$ grow?

• $a_n/n^2\to 1/4$, see my answer to essentially the same question here: math.stackexchange.com/questions/1485482/… – user138530 Oct 31 '15 at 2:40
• That's for the square root; here we'll have something like $a_n\sim n^\frac{\alpha}{\alpha-1}$ – Ivan Neretin Oct 31 '15 at 12:17
• @IvanNeretin can you show this below? – T.... Oct 31 '15 at 12:58
• Well, the main idea is the same as in Christian's answer to the other question, and the rest is just too boring. – Ivan Neretin Oct 31 '15 at 16:00