**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.