A community for students.
Here's the question you clicked on:
 0 viewing
moneybird
 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.
moneybird
 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.

This Question is Closed

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 :)
Ask your own question
Sign UpFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.