Number Sequence Finder

A number sequence is an ordered list of numbers that follow a specific rule. The two most important sequence types are arithmetic (add a constant each step) and geometric (multiply by a constant each step). Recognizing which type you have unlocks a formula to find any term or sum instantly.

Arithmetic Sequence: Each term increases by a fixed common difference (d). nth term: aₙ = a₁ + (n − 1) × d Sum of n terms: Sₙ = n/2 × (a₁ + aₙ) = n/2 × (2a₁ + (n−1)d)

Geometric Sequence: Each term is multiplied by a fixed common ratio (r). nth term: aₙ = a₁ × r^(n−1) Sum of n terms: Sₙ = a₁ × (1 − rⁿ) / (1 − r) when r ≠ 1

Worked example — Arithmetic: Sequence: 5, 11, 17, 23, 29 … (a₁ = 5, d = 6)

• Find the 20th term: a₂₀ = 5 + (20−1) × 6 = 5 + 114 = 119
• Sum of first 20 terms: S₂₀ = 20/2 × (5 + 119) = 10 × 124 = 1,240

Worked example — Geometric: Sequence: 2, 6, 18, 54, 162 … (a₁ = 2, r = 3)

• Find the 8th term: a₈ = 2 × 3^(8−1) = 2 × 2,187 = 4,374
• Sum of first 5 terms: S₅ = 2 × (1 − 3⁵) / (1 − 3) = 2 × (−242) / (−2) = 242

How to identify the type:

• Subtract consecutive terms → constant difference? Arithmetic
• Divide consecutive terms → constant ratio? Geometric
• Neither? Could be quadratic (n²), Fibonacci, or another pattern

Real-world examples:

• Arithmetic: Saving $200 more each month; stair step heights; parking meters
• Geometric: Compound interest; bacterial doubling; population growth; radioactive decay