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moneybird
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.
|dw:1322004729461:dw|
Determine the general equation of ink length for Figure n.
thinking... looks like at each step you add 5n and remove 2(n-1)
This question is from today's Canadian Intermediate Mathematics Contest
all sides are equal of that pentagon or not?
yes it's a regular pentagon
sorry I am not familiar with terminology in english
ok, I think it is:\[\frac{5n(n+1)}{2}-n(1-n)\]
sorry - I think it should be "+" after the fraction
\[\frac{5n(n+1)}{2}+n(1-n)\]
My answer is \[5+2(n-1)+\frac{3(n+2)(n-1)}{2}\]
mine simplifies to:\[\frac{n(3n+7)}{2}\]
it is basically the sum of two series: 1) 5, 5+10, 5+10+15, ... 2) 0, 0-2, 0-2-4, ...
it matches my initial thoughts on adding 5n and removing 2(n-1) after each term. interesting problem.
because at each step you are adding a new regular pentagon where each side has length n. so 5 sides makes 5n.
and every time you add a new pentagon, you cover up 2 of the previous pentagons sides - hence -2(n-1)
|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
My approach is f(n) = f(n-1) + 3n + 2
I would like to edit my answer but can't so to make it clearer we can see that \[a_1=1+1+1=3\] \[a_2=2+2+2=6\] \[d=a_2-a_1=3\]
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)
5+2(n-1) + 3/2 (n+2)(n-1)
@moneybird - your answer also simplifies to the same result :-)
yeah all resulst are equivalent :D
yeah so i got it correct on the contest!
we're ALL geniuses! :=)
what grade contest is it?
I am still in Grade 10?
I guess even in mathematics - "all roads lead to Rome"!
LOL I like that quote
thanks for posing the question @moneybird - I needed some food for my brain before going to bed :-)