Arithmetic and Geometric Sequence Calculator
Solve arithmetic and geometric sequence nth terms, generated terms, finite sums, and convergent infinite geometric sums with substituted formulas.
Compute nth terms and sums for arithmetic and geometric sequences with formula substitutions and edge-case handling.
Example/result panel
Example: with a1 = 3, d = 4, and n = 6, the arithmetic nth term is a6 = 23.
Sequence Formulas and Examples
Arithmetic sequences
Arithmetic sequences add a constant difference d each step. Nth term: a_n = a_1 + (n - 1)d. Finite sum: S_n = (n / 2) * [2a_1 + (n - 1)d].
Geometric sequences
Geometric sequences multiply by a constant ratio r. Nth term: a_n = a_1 * r^(n - 1). Finite sum: S_n = a_1 * (1 - r^n) / (1 - r) for r != 1.
Infinite geometric sums
An infinite geometric sum converges only when |r| < 1, where S_infinity = a_1 / (1 - r). If |r| >= 1, the series is non-convergent.
Edge cases
- r = 0 gives terms after the first term equal to 0.
- r = 1 gives finite sum S_n = n * a_1.
- Negative r causes alternating signs.
- Invalid n values are rejected with validation messages.
Examples and FAQ
- Arithmetic example: a_1 = 3, d = 4, n = 6 gives a_6 = 23.
- Geometric example: a_1 = 5, r = -2, n = 5 gives a_5 = 80.
- Infinite sum example: a_1 = 8, r = 0.5 gives S_infinity = 16.
Related tools: standard deviation calculator and mean, median, and mode calculator.