A Visual Proof for Geometric Series

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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

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A Visual Proof for Geometric Series

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