geometric figure illustrating the geometric progression 1,r,r2,r3,r4,r5,...

Question

Here is a geometric figure illustrating the geometric progression 1,r,r2,r3,r4,r5,…:

What is happening here? With a being the unknown side of the 2nd (=1-r-triangle) and b the unknown side of the third triangle, I know 1/b=a/r=r/x. How do I know that x=r squared? I also know that a/r=b/x. Does this help me? Do I need Pythagoras?

Also, what does this illustration proof?

score 1 · Accepted Answer · answered Apr 13 '20 at 14:47

1

We consider the triangles with hypotenuses equal $r$ and $r^2$. We let $r'$ the lenght of the second hypotenuse and we want to show that $r'=r^2$. Because the two triangles above are similar, we have: $\frac{1}{r}=\frac{r}{r'}$ So, we have: $$r'=r\cdot r=r^2$$ As shown here, we can show that the sequence $r_1,r_2,\cdots,r_n$ is a geometric progression with reason $r$, so: $$r_n=r^n, n\geq0, n \in N$$

answered Apr 13 '20 at 14:47

Matteo

6,581

Thanks. I got nowhere because I was comparing the wrong sides. – youdontknowme Apr 13 '20 at 15:59
@youdontknowme: if this answer has been useful, maybe a $+1$ or a favourite answer? – Matteo Apr 13 '20 at 20:33
Sure thing. I got confused by the fact that the second triangle needs to be flipped vertically (along the x-axis). – youdontknowme Apr 14 '20 at 12:03
@youdontknowme: are my explaination clear? – Matteo Apr 14 '20 at 12:04
Yep, very clear. r is not called "reason", though, but "ratio"'. Thanks again and stay strong, Bergamo. – youdontknowme Apr 14 '20 at 12:16

geometric figure illustrating the geometric progression 1,r,r2,r3,r4,r5,...

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