Cube as Difference between Two Squares
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 It has been suggested that this page or section be merged into Cube Number as Difference between Squares of Triangular Numbers. (Discuss) |
Theorem
A cube number can be expressed as the difference between two squares.
Proof
| \(\displaystyle n^3\) | \(=\) | \(\displaystyle \sum_{k \mathop = 1}^n k^3 - \sum_{k \mathop = 1}^{n - 1} k^3\) | |||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \paren {\frac {n^2 \paren {n + 1}^2} 4} - \paren {\frac {\paren {n - 1}^2 n^2} 4}\) | Sum of Sequence of Cubes | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \frac {n^2 \paren {\paren {n + 1}^2 - \paren {n - 1}^2} } 4\) | |||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \paren {\frac {n \paren {n + 1} } 2}^2 - \paren {\frac {n \paren {n - 1} } 2}^2\) |
$\blacksquare$
Sources
- 1980: David M. Burton: Elementary Number Theory (revised ed.) ... (previous) ... (next): Chapter $1$: Some Preliminary Considerations: $1.1$ Mathematical Induction: Problems $1.1$: $4$
