Food for thought

## Problem

What fraction of the large square is each shaded square? Write the total shaded area as a sum of fractions.

What fraction of the large square is each shaded triangle? Write the total shaded area as a sum of fractions.

If the area of the large squares is 1, what is the total sum of the shaded area in each case?

Can other sums of fractions be thought about in a similar way? Is it possible to draw a diagram that represents the following sums?

$\dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{8}+ \dfrac{1}{16}+ \dotsb$ $\dfrac{1}{3} + \dfrac{1}{9} + \dfrac{1}{27} + \dfrac{1}{81} + \dotsb$

What is the total of each sum?