If  $\mathrm{A}\left(\mathrm{a},\mathrm{b}\right)$ and  $\mathrm{B}\left(0,0\right)$ are two fixed points. ${\mathrm{M}}_1$  is the midpoint of $\overline{\mathrm{AB}},{\mathrm{M}}_2$  is the midpoint of $\overline{{\mathrm{AM}}_1},{\mathrm{M}}_3$  is the midpoint of $\overline{{\mathrm{AM}}_2}$ and so on, then ${\mathrm{M}}_5$  is

$$\begin{gathered} \mathrm{A}=(\mathrm{a},\mathrm{b}),\mathrm{B}=(0,0) \\ {\mathrm{M}}_1=\text{ midpoint of }\mathrm{AB}=\left(\frac{{\mathrm{x}}_1+{\mathrm{x}}_2}{2},\frac{{\mathrm{y}}_1+{\mathrm{y}}_2}{2}\right) \\ {\mathrm{M}}_2=\text{ midpoint of }{\mathrm{AM}}_1=\left(\frac{3{\mathrm{x}}_1+{\mathrm{x}}_2}{4},\frac{3{\mathrm{y}}_1+{\mathrm{y}}_2}{4}\right) \\ {\mathrm{M}}_5=\text{ midpoint of }=\left(\frac{31{\mathrm{x}}_1+{\mathrm{x}}_2}{32},\frac{31{\mathrm{y}}_1+{\mathrm{y}}_2}{32}\right) \end{gathered}$$
$\text{=}\left(\frac{31\left(\mathrm{a}\right)+0}{32},\frac{31\left(\mathrm{b}\right)+0}{32}\right)$$=\left(\frac{31\mathrm{a}}{32},\frac{31\mathrm{b}}{32}\right)$
(or)
If $\mathrm{A}\left(\mathrm{a},\mathrm{b}\right)$  and $\mathrm{B}\left(0,0\right)$  are two fixed points. ${\mathrm{M}}_1$  is the midpoint of $\overline{\mathrm{AB}},{\mathrm{M}}_2$  is the midpoint of  $\overline{{\mathrm{AM}}_1},{\mathrm{M}}_3$ is the midpoint of  $\overline{{\mathrm{AM}}_2}$ and so on, then 
${\mathrm{M}}_{\mathrm{n}}=\left[\left(\frac{2^{\mathrm{n}}-1}{2^{\mathrm{n}}}\right)\mathrm{a},\left(\frac{2^{\mathrm{n}}-1}{2^{\mathrm{n}}}\right)\mathrm{b}\right]$