Expert: Paul Klarreich - 2/18/2007

Question
A pyramid is made by piling up centimetre cubes layer upon layer. Each layer contains tightly packed cubes. The bottom layer is to be a square of side length 100cm and the next layer will be a square of side length 99cm and so on until there is a layer with side length 1cm (i.e. one cube), how many cubes will be needed to build the pyramid. How would you create an equation to find that out by entering the number of layers in the pyramid?

Answer
Questioner:   Stuart
Category:  Advanced Math
 
Subject:  A pile of cubes
Question:  A pyramid is made by piling up centimetre cubes layer upon layer. Each layer contains tightly packed cubes. The bottom layer is to be a square of side length 100cm and the next layer will be a square of side length 99cm and so on until there is a layer with side length 1cm (i.e. one cube), how many cubes will be needed to build the pyramid. How would you create an equation to find that out by entering the number of layers in the pyramid?
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Hi, Stuart,

Your TOP   layer has  1^2 = 1   cube.
Your next             2^2       cubes.
Your next             3^2       cubes.
.....
Your last             k^2       cubes.

Your pyramid has  1^2 + 2^2 + 3^2 + ... + k^2  cubes in it.

There is a well-known formula for the sum of  k  squares.  It's:
k(k + 1)(2k + 1)
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      6
That should do it for you.