The pirates wonder how many cannonballs would be required to build a pyramid 15 layers high (thus breaking the world cannonball stacking record). Can you help?
Let \(P(n)\) denote the number of cannonballs needed to create a pyramid \(n\) layers high. So \(P(1) = 1\text{,}\) \(P(2) = 4\text{,}\) and so on. Calculate \(P(3)\text{,}\) \(P(4)\) and \(P(5)\text{.}\)
Use polynomial fitting to find a closed formula for \(P(n)\text{.}\) Show your work.
Answer the pirate's question: how many cannonballs do they need to make a pyramid 15 layers high?
Bonus: Locate this sequence in Pascal's triangle. Why does that make sense?