Factorial Sequence

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Values of n! = n × (n−1) × ... × 2 × 1 that grow extremely quickly.

The factorial of a non-negative integer n is n! = n × (n−1) × (n−2) × ... × 2 × 1, with the convention 0! = 1. The sequence begins 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880. Each term is obtained by multiplying the previous term by the next integer, so the ratio a_{n+1}/a_n increases without bound. Factorials grow faster than any polynomial or geometric sequence.

n! = n × (n−1) × ... × 2 × 1, with 0! = 1; a_{n+1} = (n+1) × a_n

• 1, 2, 6, 24, 120 → next is 720 (multiply by 6)
• 6, 24, 120, 720 → next is 5040 (multiply by 7)
• 24, 120, 720, 5040, 40320 → ratios 5, 6, 7, 8 (successive integers)

How to recognize it

• Check the ratio between consecutive terms: if it increases by 1 each step, it is a factorial
• Factorials match 1, 2, 6, 24, 120, 720, 5040 — memorize the first seven
• Growth is faster than any geometric sequence — faster than doubling, tripling, etc.

Common mistakes

• Confusing with geometric sequences (constant ratio) when early ratios look similar
• Forgetting 0! = 1 (by convention, not by formula)
• Multiplying by n instead of (n+1) at the next step

Practice guidance

Number Sequences