Question

What are pyramidal sequences? How do they work, and can I have some examples of ones? What are their recursive and explicit formulas, and how do solve them with and without knowing the number of layers?

For context, I was faced with a problem earlier. The issue with it is that I knew how many balls were in the pyramid(300) but not how many layers, I had no clue how to solve it.

the problem goes as such:

As a project for his high school's science fair, Miguel wants to construct a stack of ping-pong balls in the shape of an ancient pyramid. For a budget of $30, he can purchase 300 balls. how many layers are in the tallest pyramid he can build?(He will have some balls left over.)​

Answer 1

• See below

Explanation:

This is about the arithmetic progression.

The sequence will be:

• 1, 3, 5, 7, ...

The common difference here is d= 2

Recursive formula:

• a₁ = 1, aₙ = aₙ₋₁ + 2

Explicit formula:

• aₙ = 1 + (n - 1)*2 = n - 1

Sum of the first n terms:

• Sₙ = (a₁ + aₙ)*n/2 = (2a₁ + (n - 1)d)*n/2 = (2 + 2n - 2)*n/2 = 2n*n/2 = n²

We need to find the nearest perfect square smaller than 300.

This is 289 = 17².

There will be 17 layers and 11 balls left over.