AI Math Assistant
BetaAsk questions about your calculation results
3 free questions per session
AI provides general assistance. Always verify important calculations.
About This Calculator
Number sequences follow specific patterns governed by rules like constant differences, constant ratios, or recursive formulas. This calculator identifies whether a sequence is arithmetic, geometric, or follows another pattern, then predicts subsequent terms and computes partial sums. Recognizing sequence patterns is a foundational skill in mathematics that extends to series analysis, financial projections, and algorithm design.
Quick Tips
- 1 Identify the pattern first — check for constant differences, then ratios.
- 2 Fibonacci-type sequences add the two previous terms to get the next one.
- 3 Geometric sequences multiply by a constant; arithmetic ones add a constant.
Example Calculation
Arithmetic sequence: 7, 13, 19, 25, 31 — find next 3 terms and sum.
Common difference: 6 | Next terms: 37, 43, 49 | Sum of first 8: 252 | Formula: 6n + 1
Arithmetic vs Geometric Sequences
An arithmetic sequence adds a constant difference between terms (2, 5, 8, 11 — difference of 3). A geometric sequence multiplies by a constant ratio (2, 6, 18, 54 — ratio of 3). This calculator generates either type and computes the sum of all terms.
Arithmetic Sequence Formulas
The nth term: aₙ = a₁ + (n-1)d. The sum of n terms: Sₙ = n/2 × (2a₁ + (n-1)d) or Sₙ = n/2 × (a₁ + aₙ). These formulas allow you to find any term or the total sum without listing all terms.
Geometric Sequence Formulas
The nth term: aₙ = a₁ × r^(n-1). The sum of n terms: Sₙ = a₁ × (1 - rⁿ) / (1 - r) when r ≠ 1. For infinite geometric series with |r| < 1, the sum converges to S = a₁ / (1 - r). This formula underlies compound interest and present value calculations.
Sequences in Mathematics and Science
Arithmetic sequences model linear growth (constant salary increases). Geometric sequences model exponential growth (compound interest, population growth) and decay (depreciation). The Fibonacci sequence, while neither arithmetic nor geometric, appears throughout nature in spiral patterns and plant growth.
Frequently Asked Questions
Check the differences between consecutive terms. If the difference is constant, it is arithmetic. If the ratio between consecutive terms is constant, it is geometric. A sequence can be neither (e.g., 1, 1, 2, 3, 5, 8 — Fibonacci).
The calculator prioritizes the common difference (arithmetic). Set the common difference to 0 to use the geometric sequence with the common ratio instead.
Yes. A negative ratio means the terms alternate in sign. For example, first term = 1, ratio = -2 gives: 1, -2, 4, -8, 16, -32, ...
The terms get progressively smaller, approaching zero. The infinite sum converges to a₁ / (1-r). For example, 1 + 1/2 + 1/4 + 1/8 + ... converges to 2.