Represent $3.\overline{42}$ on the number line, using successive magnification, up to $4$ decimal places.

To represent $3.\overline{42}$ up to $4$ decimal places, we need to locate $3.4242$ on the number line.
Step $1$: $3.4242$ lies between $3$ and $4$. Divide the interval $[3, 4]$ into $10$ equal parts and magnify the interval $[3.4, 3.5]$.
Step $2$: $3.4242$ lies between $3.4$ and $3.5$. Divide the interval $[3.4, 3.5]$ into $10$ equal parts and magnify the interval $[3.42, 3.43]$.
Step $3$: $3.4242$ lies between $3.42$ and $3.43$. Divide the interval $[3.42, 3.43]$ into $10$ equal parts and magnify the interval $[3.424, 3.425]$.
Step $4$: $3.4242$ lies between $3.424$ and $3.425$. Divide the interval $[3.424, 3.425]$ into $10$ equal parts. The point $3.4242$ is the $2$nd mark after $3.424$.