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# A large equilateral triangle is constructed by using toothpicks to cre

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Joined: 02 Sep 2009
Posts: 59725
A large equilateral triangle is constructed by using toothpicks to cre  [#permalink]

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19 Mar 2019, 21:48
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38% (03:32) correct 63% (03:24) wrong based on 8 sessions

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A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure, we have 3 rows of small congruent equilateral triangles, with 5 small triangles in the base row. How many toothpicks would be needed to construct a large equilateral triangle if the base row of the triangle consists of 2003 small equilateral triangles?

(A) 1,004,004
(B) 1,005,006
(C) 1,507,509
(D) 3,015,018
(E) 6,021,018

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Re: A large equilateral triangle is constructed by using toothpicks to cre  [#permalink]

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20 Mar 2019, 00:57
The pattern is as follows:
1st row-1 triangle-3 sticks
2nd-3 trs-6 new sticks
3rd-5 trs-9 new

So for each nth row after 1st row containing r no of triangles..No of new sticks needed = 3*r-3*(r-n)...
**(This happens because the n no of inverted triangles share the same base from the above row and the sides from the adjacent upstand triangles)

The base row contains 2003 triangles..So total no of rows upto the base row= (2003-1)/2 +1=1002
So the series becomes
(3*1-3*0)+(3*3-3*1)+(3*5-3*2)....+(3*2003 - 3*1001).... 1001 since no exclusion from 1st row as mentioned above

=3*[(1+3+5+7+..+2003)-(1+2+3+...+1001)]=3*(S1-S2)

S1= 1+3+5+..+2003 = (1002/2)*(2004)=1002^2 = 1004004
S2= 1+2+3+...+1001=1001*1002/2=1001*501 = 501501

HEnce Answr = 3*(1004004-501501) = 3*502503 =1507509.... Ans (C)
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A large equilateral triangle is constructed by using toothpicks to cre  [#permalink]

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20 Mar 2019, 02:17
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Bunuel wrote:

A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure, we have 3 rows of small congruent equilateral triangles, with 5 small triangles in the base row. How many toothpicks would be needed to construct a large equilateral triangle if the base row of the triangle consists of 2003 small equilateral triangles?

(A) 1,004,004
(B) 1,005,006
(C) 1,507,509
(D) 3,015,018
(E) 6,021,018

Attachment:
The attachment 16ca1f0305d713b2b9e5a0c53658d11829856ec7.png is no longer available

As per counting of triangles shown in figure the triangle in first row = 1
triangles in second row = 3
triangles in third row = 5
Taringles in nth row = 2003
i.e. 1+2*(n-1) = 2003
i.e. n = 1002

Considering only triangles with peak vertex up

For only one equilateral triangle at the top the Toothpicks needed = 3*1

For two rows of Equilateral triangle, Toothpicks needed = 3*(1+2) where 1+2 represents triangles with peak vertex upside

For Three rows of Equilateral triangle, Toothpicks needed = 3*(1+2+3) where 1+2+3 represents triangles with peak vertex upside

i.e. triangles needed for 1002 rows = (1+2+3+----+1002)*3 = [(1/2)*1002*1003]*3 = 1,507,509

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A large equilateral triangle is constructed by using toothpicks to cre   [#permalink] 20 Mar 2019, 02:17
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