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## 'Light Blue - Dark Blue' printed from http://nrich.maths.org/

Francesca investigated this problem. She
imagined that each time the big square was split up into little
blocks that looked like the light blue ones. Then she counted how
many light blue ones there were, and how many overall. This is what
she got :

1, $\frac{2}{3}$, $\frac{4}{9}$, $\frac{8}{27}$, $\frac{16}{81}$,
$\ldots$

She noticed that
the number on top got multiplied by 2 each time, and the number on
the bottom got multiplied by 3 each time.

Some of our more advanced readers
might know that we could write this as

$\frac{2^n}{3^n}$.

Francesca also
noticed that the amount of light blue got smaller and smaller each
time. She thinks that if we could do this forever, in the end the
whole square would be dark blue.