# 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? 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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