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 3 years ago
A pattern of Figures is shown below. Figure 1 is a regular pentagon with side length 1. Figure 2 is a regular pentagon of side length 2 drawn around Figure 1 so that the two shapes share the top vertex, T, and the sides on either side of T overlap. The pattern continues so that each n>1, Figure n is a regular pentagon of side length n drawn around the previous Figure so that the two shapes share the top vertex, T, and the sides on either side of T overlap. The ink length of each Figure is the sum of the lengths of all of the line segments in the Figure.
 3 years ago
A pattern of Figures is shown below. Figure 1 is a regular pentagon with side length 1. Figure 2 is a regular pentagon of side length 2 drawn around Figure 1 so that the two shapes share the top vertex, T, and the sides on either side of T overlap. The pattern continues so that each n>1, Figure n is a regular pentagon of side length n drawn around the previous Figure so that the two shapes share the top vertex, T, and the sides on either side of T overlap. The ink length of each Figure is the sum of the lengths of all of the line segments in the Figure.

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moneybird
 3 years ago
Best ResponseYou've already chosen the best response.4dw:1322004729461:dw

moneybird
 3 years ago
Best ResponseYou've already chosen the best response.4Determine the general equation of ink length for Figure n.

asnaseer
 3 years ago
Best ResponseYou've already chosen the best response.1thinking... looks like at each step you add 5n and remove 2(n1)

moneybird
 3 years ago
Best ResponseYou've already chosen the best response.4This question is from today's Canadian Intermediate Mathematics Contest

Tomas.A
 3 years ago
Best ResponseYou've already chosen the best response.0all sides are equal of that pentagon or not?

moneybird
 3 years ago
Best ResponseYou've already chosen the best response.4yes it's a regular pentagon

Tomas.A
 3 years ago
Best ResponseYou've already chosen the best response.0sorry I am not familiar with terminology in english

asnaseer
 3 years ago
Best ResponseYou've already chosen the best response.1ok, I think it is:\[\frac{5n(n+1)}{2}n(1n)\]

asnaseer
 3 years ago
Best ResponseYou've already chosen the best response.1sorry  I think it should be "+" after the fraction

asnaseer
 3 years ago
Best ResponseYou've already chosen the best response.1\[\frac{5n(n+1)}{2}+n(1n)\]

moneybird
 3 years ago
Best ResponseYou've already chosen the best response.4My answer is \[5+2(n1)+\frac{3(n+2)(n1)}{2}\]

asnaseer
 3 years ago
Best ResponseYou've already chosen the best response.1mine simplifies to:\[\frac{n(3n+7)}{2}\]

asnaseer
 3 years ago
Best ResponseYou've already chosen the best response.1it is basically the sum of two series: 1) 5, 5+10, 5+10+15, ... 2) 0, 02, 024, ...

asnaseer
 3 years ago
Best ResponseYou've already chosen the best response.1it matches my initial thoughts on adding 5n and removing 2(n1) after each term. interesting problem.

asnaseer
 3 years ago
Best ResponseYou've already chosen the best response.1because at each step you are adding a new regular pentagon where each side has length n. so 5 sides makes 5n.

asnaseer
 3 years ago
Best ResponseYou've already chosen the best response.1and every time you add a new pentagon, you cover up 2 of the previous pentagons sides  hence 2(n1)

Tomas.A
 3 years ago
Best ResponseYou've already chosen the best response.0dw:1322005697310:dw \[a_1=3\] \[d=3\] \[S_n=\frac{2a_1+(n1)d}{2}n=\frac{6+3n3}{2}n=\frac{3n+3n^2}{2}\] \[P=2n+S_n=3n+\frac{2n+3n^2}{2}=\frac{4n+3n+3n^2}{2}=\frac{7n+3n^2}{2}\] Let's test if n=2 and answer is 13 and it's correct

moneybird
 3 years ago
Best ResponseYou've already chosen the best response.4My approach is f(n) = f(n1) + 3n + 2

Tomas.A
 3 years ago
Best ResponseYou've already chosen the best response.0I would like to edit my answer but can't so to make it clearer we can see that \[a_1=1+1+1=3\] \[a_2=2+2+2=6\] \[d=a_2a_1=3\]

moneybird
 3 years ago
Best ResponseYou've already chosen the best response.45 + 1x2 +2x3 = f(2) = 13 f(3) = 5+ 1x2 + 2x3 + 1x2 + 3x3 = 24 f(n) = 5 + 2 + 2x3 + 2 + 3x3 + 2 + 3n = 5+ 2(n1) + 3 (2+3+4+5+6...+n)

moneybird
 3 years ago
Best ResponseYou've already chosen the best response.45+2(n1) + 3/2 (n+2)(n1)

asnaseer
 3 years ago
Best ResponseYou've already chosen the best response.1@moneybird  your answer also simplifies to the same result :)

Tomas.A
 3 years ago
Best ResponseYou've already chosen the best response.0yeah all resulst are equivalent :D

moneybird
 3 years ago
Best ResponseYou've already chosen the best response.4yeah so i got it correct on the contest!

asnaseer
 3 years ago
Best ResponseYou've already chosen the best response.1we're ALL geniuses! :=)

Tomas.A
 3 years ago
Best ResponseYou've already chosen the best response.0what grade contest is it?

moneybird
 3 years ago
Best ResponseYou've already chosen the best response.4I am still in Grade 10?

asnaseer
 3 years ago
Best ResponseYou've already chosen the best response.1I guess even in mathematics  "all roads lead to Rome"!

moneybird
 3 years ago
Best ResponseYou've already chosen the best response.4LOL I like that quote

asnaseer
 3 years ago
Best ResponseYou've already chosen the best response.1thanks for posing the question @moneybird  I needed some food for my brain before going to bed :)
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