Arithmetic & Geometric Series Calculator
Find the nth term and sum of arithmetic or geometric sequences instantly. Use it when solving textbook problems, modeling financial growth, or analyzing patterns in data.
About this calculator
An arithmetic sequence increases by a fixed common difference d each step: aₙ = a₁ + (n − 1)·d. Its sum over n terms is Sₙ = n/2 · (2a₁ + (n − 1)·d). A geometric sequence multiplies by a fixed common ratio r each step: aₙ = a₁ · r^(n−1). Its partial sum is Sₙ = a₁ · (1 − rⁿ) / (1 − r) for r ≠ 1. This calculator uses the formula: nth term = first_term + (n_terms − 1) × common_diff_ratio for arithmetic sequences. Knowing whether your sequence is arithmetic (constant difference) or geometric (constant ratio) is the first step; this determines which formulas apply.
How to use
Find the 10th term and sum of the arithmetic sequence starting at 3 with common difference 4. Enter sequence_type = Arithmetic, first_term = 3, common_diff_ratio = 4, n_terms = 10. Nth term: 3 + (10 − 1) × 4 = 3 + 36 = 39. Sum: S₁₀ = 10/2 × (2×3 + 9×4) = 5 × (6 + 36) = 5 × 42 = 210. So the 10th term is 39 and the total of the first 10 terms is 210.
Frequently asked questions
What is the difference between an arithmetic and a geometric sequence?
In an arithmetic sequence, each term is obtained by adding a fixed constant (the common difference d) to the previous term, producing a linear pattern. In a geometric sequence, each term is obtained by multiplying the previous term by a fixed constant (the common ratio r), producing an exponential pattern. For example, 2, 5, 8, 11 is arithmetic (d = 3), while 2, 6, 18, 54 is geometric (r = 3). The distinction matters because their nth-term formulas and sum formulas are fundamentally different.
How do you find the sum of an infinite geometric series?
An infinite geometric series converges only when the absolute value of the common ratio is less than 1 (|r| < 1). In that case, the sum is S = a₁ / (1 − r). For example, with a₁ = 1 and r = 0.5, S = 1 / (1 − 0.5) = 2. If |r| ≥ 1, the series diverges and has no finite sum. This concept is widely used in finance (present value of perpetuities) and physics (Zeno's paradox).
When should I use a series sum formula instead of adding terms manually?
Manual addition is practical only for very short sequences, perhaps fewer than five terms. For longer sequences or when n is large (like finding the sum of the first 100 terms), the formula Sₙ = n/2 × (2a₁ + (n−1)d) for arithmetic or Sₙ = a₁(1 − rⁿ)/(1 − r) for geometric is far more efficient. These formulas also let you work symbolically — for instance, finding what n must be for the sum to exceed a target value — which is impossible by manual addition alone.