Question

Corry collects cube containers. She fills them with water and pours them into a rectangular prism. She was playing with them and discovered something interesting. She had three cube containers. One cube was 1 inch by 1 inch by 1 inch. A second cube was 2 inches by 2 inches by 2 inches and a third cube was 3 inches by 3 inches by 3 inches.She filled all three with water. Then Corry poured all three cubes into a rectangular prism filling it exactly! The rectangular prism has a square base with a side length (1+2+3 = 6) inches and the height is 1 inch.Corry wonders whether she has found a pattern. Investigate using more cube containers to fill square base prisms with heights of 1 in. Describe any conjectures you discover.

Answer 1

`\displaystyle\sum_(i=1)^n{i^3}=\left(\sum_(i=1)^n{i}\right)^2`

Explanation:

The only sum of three cubes that has a total equal to the square of the sum of their dimensions is the one Corry found.

However, we notice that ...

1³ +2³ +3³ +4³ = (1 +2 +3 +4)² = 100

If we add the next number in sequence, we get ...

100 +5³ = 225 = (10 +5)²

225 +6³ = 441 = (15 +6)²

So, it seems that the sum of the first n cubes is equal to the square of the sum of the first n natural numbers.