For the solution, click "Read More" below.

There are different ways to reason through these problems. including by using the formula for a geometric series. Here are some ideas for reasoning through the problems without using the formula.

One-third of the square is shaded.

For this square we can focus one quarter since they differ only by a rotation.

Notice that we can think of this piece as a series of L-shaped pieces that get smaller and smaller. While the L's are all different in size, they each have the same fraction shaded.

Since 1/3 of each "L" is shaded, then 1/3 of the whole square is shaded.

Problem #2

Two-thirds of the square is shaded.

Notice that two-fourths are shaded completely, and then 1/3 of the other two-fourths are each 1/3 shaded for the same reason explained above. Adding the parts we get 1/2 + (1/3 x 1/2) = 2/3

Problem #3

Two-thirds of the square is shaded.

Notice that we can break down the square into Ls where each one is 2/3 shaded:

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