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

#Geometry #CosinesGroup #ProofWithoutWords

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

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

Math	Appears 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}`	$\boxed{123}$

Comments

Sort by:

Top Newest

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...

Raj Magesh - 7 years ago

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

Sherry Sarkar - 7 years ago

It's really amazing,

Muh. Amin Widyatama - 7 years ago

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

Sherry Sarkar - 7 years ago

Nice one for the first post. Keep posting!

Muh. Amin Widyatama - 7 years ago

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

Rohitas Bansal - 7 years ago

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

shantanu mandhane - 7 years ago

Wow. Nice proof!

faraz masroor - 7 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......

TAMIL MARX SUBASH - 6 years, 9 months ago

Math	Appears 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}`	$\boxed{123}$