Perfect Squares

I really like this month’s “Math Item of the Month” from MEI

Each number in the sequence 49, 4489, 444889, 44448889 is a perfect square.
i.e. the first digits are always 4s, the next digits are 8s but always 1 fewer of them, the units digit is always 9.

Prove that the numbers are always perfect squares.

Can you find another sequence (of the form XXXXYYZ like above but different digits) that behaves in the same way?

It would be fun to fiddle around with those numbers and see if we can figure out why that is true.  I’m going to try it.  I’m going to start with prime factorization and see if I notice anything.  Comment below if you find something interesting also?

Source:  MEI Item of the Month

BADGING:  Pre-Algebra students won’t be able to prove this mathematically.  For a badge, see if you can use some of our Problem Solving techniques (from POTW) to examine why numbers in that form are always perfect squares.  Turn in your rough/scratch work and write any conclusions or ideas you have in complete sentences.

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