Generate any sequence, term by term.
Pick arithmetic, geometric or Fibonacci, set the first term and the step, and see the nth term, the running sum, the full list of terms, and how the pattern grows.
The sequence
Arithmetic: each term adds the common difference d to the one before it.
Reading your sequence
The shape of the growth
Each term plotted in order
Key numbers
Every term and running sum
Index k, the term ak, and the sum of the first k terms.
| k | Term ak | Running sum Sk |
|---|
Sequences, explained
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.
Common questions
How do I know if a sequence is arithmetic or geometric?
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).
What if I enter both a common difference and ratio?
The calculator prioritizes the common difference (arithmetic). Set the common difference to 0 to use the geometric sequence with the common ratio instead.
Can a geometric sequence have a negative ratio?
Yes. A negative ratio means the terms alternate in sign. For example, first term = 1, ratio = -2 gives: 1, -2, 4, -8, 16, -32, ...
What happens when the geometric ratio is between -1 and 1?
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.
For learning and planning only. Very large terms or long geometric sequences can exceed normal number precision; results beyond that point are approximate. Verify critical figures by hand.