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

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Comments

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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 - 6 years, 5 months ago

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

Sherry Sarkar - 6 years, 5 months ago

It's really amazing,

Muh. Amin Widyatama - 6 years, 5 months ago

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

Sherry Sarkar - 6 years, 5 months ago

Nice one for the first post. Keep posting!

Muh. Amin Widyatama - 6 years, 5 months ago

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

Rohitas Bansal - 6 years, 5 months ago

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

shantanu mandhane - 6 years, 5 months ago

Wow. Nice proof!

faraz masroor - 6 years, 4 months 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, 2 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}`	$\boxed{123}$