12. In their down time, ghost pirates enjoy stacking cannonballs in trian- gular based pyramids (aka, tetrahedrons), like those pictured here: Note, these are solid tetrahedrons, so there will be some cannon- balls obscured from view (the picture on the right has one cannonball in the back not shown in the picture, for example) 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? (a) Let P(n) denote the number of cannonballs needed to create a pyramid n layers high. So P(1) = 1, P(2) = 4, and so on. Calculate P(3), P(4) and P(5). (b) Use polynomial fitting to find a closed formula for P(n). Show your work. 166 2. SEQUENCES (c) Answer the pirate's question: how many cannonballs do they need to make a pyramid 15 layers high? (d) Bonus: Locate this sequence in Pascal's triangle. Why does that make sense?
12. In their down time, ghost pirates enjoy stacking cannonballs in trian- gular based pyramids (aka, tetrahedrons), like those pictured here: Note, these are solid tetrahedrons, so there will be some cannon- balls obscured from view (the picture on the right has one cannonball in the back not shown in the picture, for example) 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? (a) Let P(n) denote the number of cannonballs needed to create a pyramid n layers high. So P(1) = 1, P(2) = 4, and so on. Calculate P(3), P(4) and P(5). (b) Use polynomial fitting to find a closed formula for P(n). Show your work. 166 2. SEQUENCES (c) Answer the pirate's question: how many cannonballs do they need to make a pyramid 15 layers high? (d) Bonus: Locate this sequence in Pascal's triangle. Why does that make sense?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Transcribed Image Text:12. In their down time, ghost pirates enjoy stacking cannonballs in trian-
gular based pyramids (aka, tetrahedrons), like those pictured here:
Note, these are solid tetrahedrons, so there will be some cannon-
balls obscured from view (the picture on the right has one cannonball
in the back not shown in the picture, for example)
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?
(a) Let P(n) denote the number of cannonballs needed to create
a pyramid n layers high. So P(1) = 1, P(2) = 4, and so on.
Calculate P(3), P(4) and P(5).
(b) Use polynomial fitting to find a closed formula for P(n). Show
your work.
166
2. SEQUENCES
(c) Answer the pirate's question: how many cannonballs do they
need to make a pyramid 15 layers high?
(d) Bonus: Locate this sequence in Pascal's triangle. Why does that
make sense?
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