There are some geometric means between $\dfrac {1}{2}$ and $16$. If the third mean be $4$, find the number of means between two numbers. Also find the last mean.
My Attempt Let $N$ be the number of geometric means between $\dfrac {1}{2}$ and $16$. $$a=\dfrac {1}{2}$$ $$b=16$$ Now, $$r=(\dfrac {b}{a})^{\dfrac {1}{N+1}}$$ $$r=2^{\dfrac {5}{N+1}}$$
Then, What's next!?
The means will be $\dfrac 12r, \dfrac 12r^2, \dfrac 12r^3, \dots, \dfrac 12r^n$ for some $n$. If the third mean is $4$, then
\begin{align} \dfrac 12r^3 &= 4\\ r^3 &= 8 \\ r &= 2 \end{align}
So the means are $1,2,4,8$