Cube Numbers

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Numbers of the form n³, produced by multiplying an integer by itself three times.

A cubic number (or perfect cube) has the form C_n = n³ where n is a non-negative integer. The sequence begins 0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000. Cubes grow much faster than squares: the ratio between consecutive terms approaches 1 only slowly, so cube sequences stretch out quickly. The third differences of a cube sequence are constant (equal to 6).

C_n = n³

• 1, 8, 27, 64, 125 → next is 216 (6³)
• 0, 1, 8, 27, 64 → next is 125 (5³)
• 8, 27, 64, 125, 216 → ratios 3.375, 2.37, 1.95, 1.73 show rapid growth

How to recognize it

• Check if each term is a perfect cube of an integer (1, 8, 27, 64, 125, 216, ...)
• Take differences three times; if the third differences are constant at 6, you have cubes
• Cubes grow much faster than squares — if values explode quickly, suspect n³

Common mistakes

• Confusing cubes with squares when the first term is 1 (both sequences start with 1)
• Mistaking 2n³ or n³ + 1 for a pure cube sequence
• Assuming a geometric sequence when the growth is actually polynomial

Practice guidance

Number Sequences