A question from Ramaiah entrance test

Ramaiah's classes were (are) popular coaching classes for IIT JEE exam preparation in Hyderabad. They themselves had an entrance exam!

These Ramaiah entrance tests contained some very nice problems.

This was a question I first saw on a Ramaiah entrance test when my brother appeared for it (1990). The question is actually much older than that, probably the 1800s.

Anyway, here goes:

Consider the sequence of numbers

$$49, 4489, 444889, \dots$$

Each number is formed by inserting 48 in the middle of the previous number.

$$49 \to 4\underline{48}9 \to 44\underline{48}89 \to \dots$$

Show that all these numbers are perfect squares!

[Solution]