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Isosceles triangle 
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#1
Jun2008, 03:36 PM

P: 908

1. The problem statement, all variables and given/known data
Hi! After a tiring excursion, finally I am back... Here is one for you: In isosceles triangle ABC (triangle with all sides equal size) with side a, is drawn another isosceles triangle [itex]A_1,A_2,A_3[/itex] which points [itex]A_1,A_2,A_3[/itex] are in AB/2, BC/2, AC/2. Again third triangle with same attributes is drawn, fourth, fifth.... infinite... This looks like on this picture. Find the sum of the perimeter and calculate the sum of the areas of the triangles. 2. Relevant equations 3. The attempt at a solution I think it is something like this: [tex]P + \frac{P}{4} + \frac{P}{8} + ... + \frac{P}{2^n}[/tex] for the area of the triangle, and [tex]L + \frac{L}{2} + \frac{L}{4} + ... + \frac{L}{2^n^^1}[/tex] for the perimeter. I think also, that I can write them as: [tex] P + \sum_{n=2}^n \frac{P}{2^n} = P + \frac{P}{4} + \frac{P}{8} + ... + \frac{P}{2^n} [/tex] [tex] L + \sum_{n=2}^n \frac{L}{{2}^{n1}} = L + \frac{L}{2} + \frac{L}{4} + ... + \frac{L}{{2}^{n1}} [/tex] [tex]n \in \mathbb{N}[/tex] [tex]n\geq 2[/tex] n  number of triangles 


#2
Jun2008, 03:48 PM

P: 677

Do you know what a geometric sum is?



#3
Jun2008, 03:55 PM

P: 908

Yes, I write the geometric sums above in the first post. Please see them, now.



#4
Jun2008, 05:44 PM

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P: 26,148

Isosceles triangle
Hint: if the lengths are halved each time, what happens to the area? And what is ∑x^n? (oh, and they're actually equilateral … isoceles means two sides equal) 


#5
Jun2108, 03:14 AM

P: 908

tinytim I was on excursion, Skopje  Belgrade  Bratislava  Prague  Vienna (approximately 3000 km in both ways)
Is this correct: [tex] P + \sum_{k=2}^n \frac{P}{2^k} = P + \frac{P}{4} + \frac{P}{8} + ... + \frac{P}{2^k} [/tex] [tex] L + \sum_{k=2}^n \frac{L}{{2}^{k1}} = L + \frac{L}{2} + \frac{L}{4} + ... + \frac{L}{{2}^{k1}} [/tex] [tex] n \in \mathbb{N} [/tex] [tex] n\geq 2 [/tex] n  number of triangles 


#6
Jun2108, 05:52 AM

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Hi Physicsissuef!
The L equation is correct, the P equation isn't. Hint, repeated: if the lengths are halved each time, what happens to the area? 


#7
Jun2108, 06:01 AM

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P: 12,069

The area equation is incorrect, starting with the term P/8. P and P/4 are correct.
Hint: think carefully, what is the area of the 3rd triangle? (It's not P/8) 


#8
Jun2108, 06:25 AM

P: 908

Ahh... I understand. Maybe this is better:
[tex] \sum_{k=0}^n \frac{P}{4^k} = P + \frac{P}{4} + \frac{P}{16} + ... + \frac{P}{4^k} [/tex] [tex]n \in \mathbb{N}[/tex] [tex]n \geq 2[/tex] n number of triangles. Also for L: [tex] \sum_{k=0}^n \frac{L}{2^k} = L + \frac{L}{2} + \frac{L}{4} + ... + \frac{L}{2^k} [/tex] 


#9
Jun2108, 06:34 AM

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ok, now what is: [tex]\sum_{k=0}^n \left(\frac{1}{x}\right)^k[/tex] ? 


#10
Jun2108, 08:50 AM

P: 908

It will work also with "k". I sow on wikipedia.
[tex] \sum_{k=0}^n \left(\frac{1}{x}\right)^k = 1 + \frac{1}{x} + \frac{1}{x^2} + ... + \frac{1}{x^n} [/tex] Why 1/x ? 


#11
Jun2108, 10:10 AM

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#12
Jun2108, 11:02 AM

P: 908

[tex]\frac{\frac{1}{x}\frac{1}{{x}^{n+1}}}{1\frac{1}{x}}}[/tex]



#13
Jun2108, 11:06 AM

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#14
Jun2108, 12:31 PM

P: 908

But isn't something like this:
[tex]S=d+d^2+...+d^n[/tex] [tex]Sd=d^2+d^3+...+d^n^+^1[/tex] [tex]SSd=S(1d)=dd^n^+^1[/tex] [tex]S=\frac{dd^n^+^1}{1d}[/tex] ? 


#15
Jun2108, 12:57 PM

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P: 2,616

Your first term is 1, not 1/x. The formula you derived needs to be modified slightly to suit your problem.



#16
Jun2108, 01:22 PM

P: 908

if d=1 then also down there it will be 1..
[tex] S=\frac{1d^n^+^1}{11} [/tex] 


#17
Jun2108, 01:26 PM

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P: 2,616

No, in your derivation, the first term is equivalent to r, the ratio of the n+1 term to the nth term. But in [tex]\sum_{k=0}^n \left(\frac{1}{x}\right)^k = 1 + \frac{1}{x} + \frac{1}{x^2} + ... + \frac{1}{x^n}[/tex], the first term is not equivalent to the ratio. Which is why the formula needs to be modifed in order to suit your problem.



#18
Jun2108, 01:27 PM

P: 908

[tex]S = 1 + \frac{1}{x} + \frac{1}{x^2} + ... + \frac{1}{x^n} [/tex]
[tex]S\frac{1}{x}=\frac{1}{x} + \frac{1}{x^2} + \frac{1}{x^3} + ...+ \frac{1}{{x}^{n+1}}[/tex] [tex]SS\frac{1}{x}=S(1\frac{1}{x})=1\frac{1}{{x}^{n+1}}[/tex] [tex]S=\frac{1\frac{1}{{x}^{n+1}}}{1\frac{1}{x}}[/tex] Sorry, you're right. But I need to do the same for P and L, right? 


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