# 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
6 years ago

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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...

- 6 years ago

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

- 6 years ago

It's really amazing,

- 6 years ago

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

- 6 years ago

Nice one for the first post. Keep posting!

- 6 years ago

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

- 6 years ago

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

- 6 years ago

Wow. Nice proof!

- 6 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......

- 5 years, 10 months ago