Let's put a little geometry into geometric series! The spiral in the picture is made from 45° right triangles. The smallest triangle (red) has a side length of
1. You can use the Pythagorean theorem to find that the length of its
hypotenuse is 2. The next (orange) triangle has a side length exactly
matching the hypotenuse of the smallest triangle (which was just mentioned
as being 2). The next (yellow) triangle has a side length of 2; the next
triangle has a side length of 22, and so on.
The areas of the colored triangles, calculated with the formula
2.
• base • height, are 3. 1.2.4, and so on. What is the area of this entire
colorful figure?