# A Visual Proof for Geometric Series

I personally love the idea of Proofs Without Words! I like how simple, elegant and creative they can be. So here is another one for all you proof enthusiasts. Link

Expansion

In this proof, we see that $$\frac{1}{1-r}$$ = $$1 + r + r^2 + r^3 ...$$.

$$\triangle TPS$$ is cut into multiple similar triangles with a ratio r. $${ST}$$ is $$1 + r + r^2 + r^3 ...$$. For the purposes of this proof, let the point where line $${QR}$$ meets $$ST$$ be $$A$$.

$$\angle PRQ = \angle TRA$$ because they're vertical angles. $$\angle PQR = \angle TAR$$ since they are both right angles. Thus $$\triangle PRQ \sim \triangle TRA$$ by $$AA$$ Similarity. Since $$\triangle TPS \sim \triangle TRA$$, $$\triangle PRQ \sim \triangle TPS$$.

Tinkering with our ratios, we get $$\frac{PQ}{QR} = \frac{TS}{SP}$$

$$\frac{1}{1-r} = \frac{1 + r + r^2 + r^3 ...}{1}$$

Q.E.D

Note by Sherry Sarkar
5 years ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

Sort by:

This is awesome! Although perhaps it is worth noting that $$|r| < 1$$ for this to work. I wonder if I could think up a similar geometric proof for the more general formula: $$1 + r + r^{2} +... = \dfrac{1-r^{n}}{1-r}$$... Hmm...

- 5 years ago

Thank you for the feedback, I forgot to add that detail!

- 5 years ago

It's really amazing,

- 5 years ago

Thanks! It's my first post, so I'm on the lines here. :D

- 5 years ago

Nice one for the first post. Keep posting!

- 5 years ago

Elegant Maths... that's what i call it

- 5 years ago

that's fantastic!!!!!!!!!

- 5 years ago

Wow. Nice proof!

- 5 years ago

We will get the same answer if we extend the dotted triangle towards the right:

If we extend upto r^2, we get $(1+r)/(1- r^{2} )$ which can be simplified to get the same answer.

If we extend upto r^3, we get $(1+r+r^{2})/(1-r^{3})$ which again can be simplified to get the same result. And so on......

- 4 years, 10 months ago

×