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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- anonymous

|dw:1322004729461:dw|

- anonymous

Determine the general equation of ink length for Figure n.

- asnaseer

thinking...
looks like at each step you add 5n and remove 2(n-1)

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## More answers

- anonymous

This question is from today's Canadian Intermediate Mathematics Contest

- anonymous

all sides are equal of that pentagon or not?

- anonymous

yes it's a regular pentagon

- anonymous

sorry I am not familiar with terminology in english

- asnaseer

ok, I think it is:\[\frac{5n(n+1)}{2}-n(1-n)\]

- asnaseer

sorry - I think it should be "+" after the fraction

- asnaseer

\[\frac{5n(n+1)}{2}+n(1-n)\]

- anonymous

My answer is
\[5+2(n-1)+\frac{3(n+2)(n-1)}{2}\]

- asnaseer

mine simplifies to:\[\frac{n(3n+7)}{2}\]

- asnaseer

it is basically the sum of two series:
1) 5, 5+10, 5+10+15, ...
2) 0, 0-2, 0-2-4, ...

- asnaseer

it matches my initial thoughts on adding 5n and removing 2(n-1) after each term.
interesting problem.

- anonymous

Why is it 5n?

- asnaseer

because at each step you are adding a new regular pentagon where each side has length n. so 5 sides makes 5n.

- asnaseer

and every time you add a new pentagon, you cover up 2 of the previous pentagons sides - hence -2(n-1)

- anonymous

|dw:1322005697310:dw|
\[a_1=3\]
\[d=3\]
\[S_n=\frac{2a_1+(n-1)d}{2}n=\frac{6+3n-3}{2}n=\frac{3n+3n^2}{2}\]
\[P=2n+S_n=3n+\frac{2n+3n^2}{2}=\frac{4n+3n+3n^2}{2}=\frac{7n+3n^2}{2}\]
Let's test if n=2 and answer is 13 and it's correct

- anonymous

My approach is
f(n) = f(n-1) + 3n + 2

- anonymous

I would like to edit my answer but can't so to make it clearer we can see that
\[a_1=1+1+1=3\]
\[a_2=2+2+2=6\]
\[d=a_2-a_1=3\]

- anonymous

5 + 1x2 +2x3 = f(2) = 13
f(3) = 5+ 1x2 + 2x3 + 1x2 + 3x3 = 24
f(n) = 5 + 2 + 2x3 + 2 + 3x3 + 2 + 3n
= 5+ 2(n-1) + 3 (2+3+4+5+6...+n)

- anonymous

5+2(n-1) + 3/2 (n+2)(n-1)

- asnaseer

@moneybird - your answer also simplifies to the same result :-)

- anonymous

yeah all resulst are equivalent :D

- anonymous

yeah so i got it correct on the contest!

- asnaseer

we're ALL geniuses! :=)

- anonymous

what grade contest is it?

- anonymous

Grade 8,9, and 10

- anonymous

I am still in Grade 10?

- asnaseer

I guess even in mathematics - "all roads lead to Rome"!

- anonymous

LOL I like that quote

- asnaseer

thanks for posing the question @moneybird - I needed some food for my brain before going to bed :-)

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