Squares & Cubes Pattern

Examine the numerical statements in the box. Something is going on.

What would the statement be for 100³?

What would the statement be for n³?

Can you prove that it always works?

Show/Hide Solution

We observe: 4³ = 10² – 6² → 4³ = (4·5/2)² – (4·3/2)² = (4(4 + 1)/2)² – (4(4 – 1)/2)²

This pattern also holds for the other examples given.

So we try:

100³ ≟ ((100 · 101)/2)² – ((100 · 99)/2)² →
1,000,000 ≟ 25,502,500 – 24,502,500
Yes, ok.