There are some geometric means between 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 $1,2,4,8$
