# Arithmetic & Geometric Sequence Solver Generate terms, calculate general formulas, and sum series for arithmetic and geometric progressions with this free solver. --- - **Canonical URL:** https://dothecalculation.com/calculators/number-sequence-calculator - **Category:** Math calculators - **Publisher:** Do The Calculation (https://dothecalculation.com) - **Cost:** Free, no account or sign-up required - **Privacy:** Runs entirely in the browser; inputs are never sent to a server - **Methodology:** https://dothecalculation.com/methodology --- ## Arithmetic & Geometric Sequence Solver Generate sequence terms, solve general formulas, and compute series summation for arithmetic and geometric progressions. - Common differences & ratios - n-th term formula & solvers - Infinite series convergence limits ## Introduction to Number Sequences and Progressions A number sequence is an ordered list of numbers that follow a specific mathematical rule. The individual numbers in a sequence are called terms. Sequences are fundamental in algebra, calculus, and discrete mathematics, and they appear in natural patterns (like the Fibonacci sequence in plants) and financial models (like compound interest progressions). The sum of the terms of a sequence is called a series. Solving sequences involves finding the general formula for the \(n\)-th term, generating list values, and evaluating finite or infinite summations. The two most common types of sequences are arithmetic sequences and geometric sequences. In an arithmetic sequence, the difference between consecutive terms is constant. In a geometric sequence, the ratio of consecutive terms is constant. Both progressions have unique formulas for finding terms and summing series, which are essential for engineering analysis, physics models, and financial planning. This calculator generates sequences step-by-step. By entering the first term \(a_1\), the common difference \(d\) or ratio \(r\), and the number of terms \(n\) (up to 30), you can calculate any specific term, find the algebraic summation \(S_n\), and examine whether infinite geometric series converge to a finite limit. ## Arithmetic Progressions (AP) An arithmetic sequence increases or decreases by adding a constant value called the common difference \(d\). The terms follow the pattern: \(a_1, a_1 + d, a_1 + 2d, \dots\) The formula for the \(n\)-th term \(a_n\) is: $$a_n = a_1 + (n - 1)d$$ The sum of the first \(n\) terms (a finite arithmetic series) is calculated as: $$S_n = \frac{n}{2}(a_1 + a_n) = \frac{n}{2}[2a_1 + (n - 1)d]$$ This summation formula was famously derived by Carl Friedrich Gauss as a young student. ## Geometric Progressions (GP) and Infinite Series A geometric sequence increases or decreases by multiplying by a constant value called the common ratio \(r\). The terms follow the pattern: \(a_1, a_1 r, a_1 r^2, \dots\) The formula for the \(n\)-th term \(a_n\) is: $$a_n = a_1 r^{n-1}$$ The sum of the first \(n\) terms is calculated as: $$S_n = a_1 \frac{1 - r^n}{1 - r} \quad (r \neq 1)$$ If the common ratio satisfies \(|r| < 1\), the infinite geometric series converges to a finite limit as \(n \to \infty\): $$S_{\infty} = \frac{a_1}{1 - r}$$ If \(|r| \geq 1\), the infinite series diverges to infinity. ## How to Use This Calculator Choose Arithmetic or Geometric, enter the first term \(a_1\), the common difference \(d\) (arithmetic) or common ratio \(r\) (geometric), and the number of terms \(n\) to generate (capped at 30). The calculator lists every term, computes the \(n\)-th term and the sum of all \(n\) terms, and, for geometric sequences with \(|r| < 1\), the infinite-series limit. ## Worked Example: Arithmetic Sequence, a₁ = 2, d = 3, n = 6 Terms: \(2, 5, 8, 11, 14, 17\) — the calculator's default inputs. $$a_6 = a_1 + (n-1)d = 2 + (6-1)(3) = 17$$ $$S_6 = \frac{n}{2}(a_1 + a_6) = \frac{6}{2}(2 + 17) = 3 \times 19 = 57$$ So the 6th term is 17 and the sum of the first six terms is 57. ## Related Calculators For a specific recurrence sequence rather than a general arithmetic/geometric progression, see the [Fibonacci and golden ratio calculator](/calculators/fibonacci-golden-ratio-calculator). To sum an infinite series of function terms instead of a numeric progression, use the [Taylor series calculator](/calculators/taylor-series-calculator). ## Frequently asked questions ### What is a sequence? A sequence is an ordered list of numbers that follow a specific mathematical rule or pattern. ### What is an arithmetic sequence? A sequence where each term is found by adding a constant common difference (d) to the previous term. ### What is a geometric sequence? A sequence where each term is found by multiplying the previous term by a constant common ratio (r). ### How do you find the n-th term of an arithmetic sequence? Use the formula: aₙ = a₁ + (n−1)d, where a₁ is the first term and d is the common difference. ### How do you find the n-th term of a geometric sequence? Use the formula: aₙ = a₁ · r^(n−1), where a₁ is the first term and r is the common ratio. ### What is a series? A series is the sum of the terms of a sequence. ### What is the sum formula for an arithmetic series? The sum is Sₙ = (n/2)(a₁ + aₙ) or Sₙ = (n/2)[2a₁ + (n−1)d]. ### What is the sum formula for a geometric series? The sum is Sₙ = a₁(1 − rⁿ)/(1 − r) for r ≠ 1. ### When does an infinite geometric series converge? It converges if and only if the absolute value of the common ratio is strictly less than 1 (|r| < 1). ### What is the sum of an infinite convergent geometric series? The sum is S∞ = a₁ / (1 − r). ### What is the difference between convergence and divergence? Convergence means the terms or sum approach a specific finite value. Divergence means they grow infinitely larger or oscillate without limit. ### Can a common difference or common ratio be negative? Yes, a negative common difference means the arithmetic sequence decreases. A negative common ratio means the geometric sequence alternates between positive and negative values. ## Related concepts - **Infinite Series** — The sum of infinitely many terms of a sequence. - **Gauss Summation** — The method for summing consecutive integers. - **Convergence Limits** — The value that terms approach as they extend to infinity. ## Related guides - [Scientific Notation Guide: Powers of Ten Made Practical](https://dothecalculation.com/blog/math/scientific-notation-basics) — Learn how to convert, compare, and calculate with powers of ten using worked examples and the live DTC scientific calculator. ## Related calculators - [Linear Programming Solver (Graphical Method)](https://dothecalculation.com/calculators/linear-programming-calculator) — Optimize a 2-variable objective function subject to linear inequality constraints using the graphical corner-point method with this free solver. - [System of Linear & Nonlinear Equations Solver](https://dothecalculation.com/calculators/system-of-equations-calculator) — Solve a system of two linear equations via Cramer’s Rule, or a line-and-parabola nonlinear system via substitution, with clear steps and an intersection graph. - [Cross Multiplication Calculator](https://dothecalculation.com/calculators/cross-multiplication-calculator) — Use our free cross multiplication calculator to solve equations with fractions. 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