How do I get the equation for the 10th, 20th and nth members in the sequence?   The sequence is composed of triangles to make a pyramid; 1. 1 trangle; 3 matchsticks 2. 4 triangles; 9 matchsticks 3. 9 triangles; 18 matchsticks Looks like this: http://nrich.maths.org/88 How do I find the pattern to make the equation to find the 10th, 20th, and nth members of the sequence?

Expert Answers

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Look how the pyramid is constructed.

The pyramid is constructed inside an equilateral triangle of size n sticks. It is filled with small equilateral triangles of size 1.

If `a_n ` is the number of small triangles in the big triangle of side n.

`b_n` is the number of sticks needed to fill the big triangle of side n.

It build the next step, we just need to add n+1 small upward traingles of side 1

/_\ /_\  ... /_\

 

Which means we add 3(n+1) sticks.

Therefore `b_(n+1)=b_n+3(n+1)`

 

How many small triangles were added?

 (n+1) upward and n downward. i.e 2n+1 in total.

`a_(n+1)=2n+1+a_n=2(n+1)-1+a_n`

 

Let's try to find a expression of `b_n` as a function of n

`b_n=3+3*2+3*3+3*4+....+3*n=3(1+2+...+n)=3*n(n+1)/2`

 

b_n=3n(n+1)/2

 

Let's find `a_n`

`a_n=2*1-1+2*2-1+2*3-1+...2*n-1=2(1+2+3+...+n)-1-1-...-1`

`a_n=2n(n+1)/2-n=n(n+1)-n=n^2.`

 

Therefore, at the level n there are `3n(n+1)/2` sticks and `n^2 ` triangles.

 

 

 

 

 

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