# CUBIC NUMBERS | CUBES

An online cubic number definition

A cubic number is the third power of a number as in a x a x a = a^{3}.

Those familiar with the evolution of the squares from adding successive odd numbers might not be too surprised to discover how the cubes evolve from summing odd numbers also. Clearly, the nth cube is simply n^{3}. Cubes can be derived in other ways also.

The cube of any integer, “n”, is the sum of the series of odd numbers beginning with (n^{2} – n + 1) and ending with (n^{2} + n – 1).

**For example:** For n = 6, (n^{2} – n + 1) = 31 and 31 + 33 + 35 + 37 + 39 + 41 = 216 = 6^{3}

### Example

n | (n)^{3} |
= | n^{3} |
(n^{2} – n + 1) |
= | (n^{2} + n – 1) |

1 | 1^{3} |
= | 1 | 1 | = | 1(1) |

2 | 2^{3} |
= | 8 | 3+5 | = | 2(1+2+1) |

3 | 3^{3} |
= | 27 | 7+9+11 | = | 3(1+2+3+2+1) |

4 | 4^{3} |
= | 64 | 13+15+17+19 | = | 4(1+2+3+4+3+2+1) |