Sum of a Geometric Series

r

0.5625

a

1.97

All the green triangles are similar, and the big triangle is similar to the green triangles.

The ratio of the big triangle's sides is equal to the ratio of the sides for those triangles, so that

a+ar+a

2

r

+⋯+a

n

r

+⋯

1

r

=

ar

1-r

, and then

∞

∑

n=0

a

n

r

=

a

1-r

.