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Triangles are added in a clockwise direction to create the spiral of triangles shown below. If the length of the sides of the triangles in the given pattern are given by the sequence 1, 1, 1, 2, 2, 3 ..., what is the length of the side of the 20th triangle in this sequence?

 

 Jun 7, 2020
 #1
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This might be the right way to solve this problem:

 

The expression "T[6]" means triangle number 6; the number to the right is the length of each side.

 

T[1]   --->   1

T[2] = T[1]    --->   1

T[3] = T[2]    --->   1

T[4] = T[1] + T[3]   --->   2   (add the length of a side of triangle 1 to the length of the side of triangle 3)

T[5] = T[4]   --->   2

T[6] = T[5] + T[1]   --->   3   (add the length of a side of triangle 5 to the length of the side of triangle 1)

T[7] = T[6] + T[2]   --->   5

T[8] = T[7] + T[3]   --->   6

T[9] = T[8] + T[4]   --->   8

 

From now on, the pattern is:  T[n]  =  T[n - 1] + T[n - 5]   ...

 Jun 7, 2020

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