SOLUTION: GREAT PYRAMID OF ORANGES A very bored grocer was stacking oranges one day. She decided to stack them in a triangular pyramid. There was one orange in the top layer, three orang

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: GREAT PYRAMID OF ORANGES A very bored grocer was stacking oranges one day. She decided to stack them in a triangular pyramid. There was one orange in the top layer, three orang      Log On


   

Question 1161157: GREAT PYRAMID OF ORANGES
A very bored grocer was stacking oranges one day. She decided to stack them in a triangular pyramid. There was one orange in the top layer, three oranges in the second layer, six oranges in the third layer, and so on. Each layer except the top formed an equilateral triangle.
How many oranges would it take her to build a pyramid 50 layers high?
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52780) About Me  (Show Source):
You can put this solution on YOUR website!
.

I will not go into the solution.

I simply recommend you to read these Wikipedia articles, related to the subject

https://en.wikipedia.org/wiki/Triangular_number#:~:text=0%2C%201%2C%203%2C%206,%2C%20630%2C%20666...

https://en.wikipedia.org/wiki/Tetrahedral_number



Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


As suggested by tutor @ikleyn, there is a lot of fun mathematics in the patterns involved in solving this problem.

Counting from the top row, the number of oranges in the n-th layer is the n-th triangular number, which is C%28n%2B1%2C2%29

Those numbers are found in a diagonal of Pascal's Triangle.

The hockey stick identity in Pascal's Triangle (another internet search for you, if you are not familiar with it), tells us that

sum%28C%28n%2B1%2C2%29%2C1%2Cn%29 = C%28n%2B2%2C3%29

So the number of oranges in the stack of 50 layers is C%2852%2C3%29+=+22100