Determining the growth rate of a function

[DanSlevin]

I'm trying to figure out the growth rate of a function. Below is what I believe to be the solution, but I'm wondering if I've properly taken into account all the factors necessary, so I wanted to see if this appears correct.

$$\Large\frac{3(n+1)^{\frac{2}{3}}}{2}-\frac{3(1)^{\frac{2}{3}}}{2}$$

$$\Large\frac{3}{2}((n+1)^{\frac{2}{3}}-1) $$

$$\Large\Theta(n^{\frac{2}{3}}) $$

 

[CaptainBlack]

I'm trying to figure out the growth rate of a function. Below is what I believe to be the solution, but I'm wondering if I've properly taken into account all the factors necessary, so I wanted to see if this appears correct.

$$\Large\frac{3(n+1)^{\frac{2}{3}}}{2}-\frac{3(1)^{\frac{2}{3}}}{2}$$

$$\Large\frac{3}{2}((n+1)^{\frac{2}{3}}-1) $$

$$\Large\Theta(n^{\frac{2}{3}}) $$

It does not appear correct because you provide no explanation of how you get from the first line to the last or indeed what the relationship between the expression on the first line is with that on the last.

It is indeed the case that

\[\large \left[ \frac{3(n+1)^{2/3}}{2}-\frac{3(1)^{2/3}}{2}\right]\in \Theta(n^{2/3}) \]

but I won't say your explanation is inadequate because it is not an explanation at all. Also the form of your first line suggests that this is part of a larger problem, which you really should have posted.

CB