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: 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.



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





Let's find `a_n`




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






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