Problem:
Solution:
Solution:
Problem:
Solution:
Solution:
This last one follows from similarity of the subsequent trapezoids: the right edge of the teal(ish) trapezoid has length
, and so the right edge of the neighboring trapezoid,
, is found by
, and we see that it has length
.
We may come up with infinitely many proofs of these geometric series! All we need is a figure which can be dissected into self-similar parts, where the geometric series is a sum of powers of
. Awesome.