Geometric Sequence Calculator - n-th Term and Sum Finder

A geometric sequence calculator finds the n-th term, the first n terms, the finite sum, and the sum to infinity of a geometric progression, which is the sequence formed by multiplying each term by a fixed common ratio r. Type the first term, the common ratio, and the number of terms, and the calculator lists every term in order alongside the closed-form a_n, S_n, and S_infinity values for the same progression.

• • Solve homework and exam problems: Type a_1, r, and n to read the n-th term and the finite sum for textbook and competition problems without doing the arithmetic by hand.
• • Project a compounding quantity: Use a_1 as the starting amount and r = 1 + growth rate to read a_n as the value after n periods and S_n as the cumulative total.
• • Test whether an infinite sum converges: Type any r to read the sum to infinity whenever |r| < 1, and a clear divergence message when |r| >= 1.

The geometric sequence is the multiplicative counterpart of the arithmetic sequence, so it shows up anywhere a quantity grows or shrinks by a fixed ratio each step, and the calculator prints the n-th term, the finite sum, the infinite sum, and the full term list on the same screen.

Internally the calculator validates that a_1, r, and n are finite numbers with n being a positive integer, then applies the closed-form n-th term formula and the closed-form finite sum, and checks whether |r| is strictly less than 1 before returning a sum to infinity.

• a₁: The first term. Any real number, including 0 and negatives.
• r: The common ratio. Each term is the previous term times r. Infinite sum converges only when |r| < 1.
• n: Positive integer count of terms to list, and the index of a_n.
• aₙ: The n-th term, equal to a_1 * r^(n-1).
• Sₙ: Sum of first n terms. Equals a_1 * (1 - r^n) / (1 - r) when r is not 1, and n * a_1 when r = 1.
• S∞: Sum to infinity, equal to a_1 / (1 - r) whenever |r| < 1.

Wikipedia defines the n-th term of a geometric progression as a_n = a_1 * r^(n-1), and Wolfram MathWorld writes the finite sum as a_1 * (1 - r^n) / (1 - r) with the infinite sum converging to a_1 / (1 - r) whenever |r| < 1, which is exactly what the calculator prints.

According to Wikipedia (Geometric progression), the n-th term of a geometric progression with first term a_1 and common ratio r is a_n = a_1 * r^(n-1), and the sum of the first n terms is a_1 * (1 - r^n) / (1 - r) when r is not 1

According to Wolfram MathWorld (Geometric Series), the sum of the first n terms of a geometric series is a_1 * (1 - r^n) / (1 - r), and an infinite geometric series converges to a_1 / (1 - r) whenever |r| < 1

When each step adds a fixed number instead of multiplying by a fixed ratio, the Arithmetic Sequence Calculator applies the additive counterpart formula a_n = a_1 + (n-1) * d on the same input shape.

Four ideas are the only prerequisites for using a geometric sequence correctly on any input triple.

When the first term carries a unit, every term and the sum inherit that unit; when r is a growth factor like 1.08, the n-th term and the sum are the values after n periods of compounding.

When the same list of n numbers should be averaged as a multiplicative mean instead of a step-by-step progression, the Geometric Mean Calculator returns the n-th root of the product on the same comma-separated input.

Four short steps cover every workflow the geometric sequence calculator supports, from a single textbook term to a long list of compounding values.

1. 1 Type the first term a_1: Enter the first term of the sequence. Decimals and negative numbers are accepted; zero is allowed and produces a sequence of all zeros.
2. 2 Type the common ratio r: Enter the fixed multiplier between consecutive terms. A ratio of 2 doubles, a ratio of 0.5 halves, and a ratio of 1 keeps every term equal to a_1.
3. 3 Type the number of terms n: Enter the positive integer count of terms to list. The calculator caps n at 1000 to keep the term list readable.
4. 4 Read a_n, S_n, S_infinity, and the term list: The Results panel prints the n-th term, the sum of the first n terms, the sum to infinity (or a divergence note), the count of terms listed, and the comma-separated first n terms.

A savings account starts at $1,000 (a_1 = 1000) and grows by 5% each year (r = 1.05). Type those into the inputs with n = 10 to read a_10 ≈ $1,628.89 (the balance after 10 years) and S_10 ≈ $12,577.89 (the cumulative total), with the sum to infinity flagged as diverging because |r| > 1.

When the geometric sequence is modelling a real savings or loan balance, the Compound Interest Calculator turns the same a_1, r, and n into a year-by-year balance and total interest schedule.

The calculator removes the most common geometric-sequence mistakes.

• • All five answers in one pass: n-th term, finite sum, sum to infinity, term count, and the full term list print at the same time.
• • Handles the r = 1 edge case: When the common ratio is exactly 1, the calculator switches to the simplified sum S_n = n * a_1 so it never divides by zero on a constant sequence.
• • Detects divergence cleanly: When |r| >= 1 the sum to infinity does not converge, so the calculator returns a clear divergence note instead of Infinity or NaN.
• • Capped at 1000 terms for readability: A typo on the n input cannot lock the browser trying to render millions of terms.
• • Pairs with related math tools: The Arithmetic Sequence Calculator applies the additive counterpart on the same n input.

If you are working through a problem set or a financial report, the calculator removes the chance of mixing up the n-th term with the sum, and the result updates as you type so it tolerates decimals, scientific notation, and negative ratios.

When two consecutive terms should be expressed as a:b in lowest terms to read the common ratio as a fraction, the Ratio Calculator simplifies a:b and a:b:c ratios and finds a missing term in a:b = c:d.

Three things change the answer you should expect from the geometric sequence calculator, plus two practical caveats about how geometric sequences behave in the real world.

• • A geometric sequence is the right model for quantities that change by a fixed ratio each step, but it is the wrong model for quantities that change by a fixed amount. Average temperatures, test scores, and any list that adds together should use the Arithmetic Sequence Calculator instead.
• • The sum to infinity is only a meaningful answer when |r| < 1. For any r outside that range the infinite sum diverges, and the calculator returns a divergence note rather than a numeric answer.

When you copy the n-th term or the sum into a spreadsheet, double-check the unit context. If a_1 is in dollars, every term and the sum are in dollars. If r is a growth factor like 1.08, the n-th term is the value after n periods, and you subtract a_1 to read the dollar growth.

According to Wolfram MathWorld (Geometric Progression), a geometric progression is a sequence a_n where each successive term is the previous term multiplied by a fixed non-zero number called the common ratio, so a_n = a_1 * r^(n-1) with sum a_1 * (r^n - 1) / (r - 1)

When the n-th term prints in scientific notation because r^(n-1) is very large, the Exponential Notation Calculator reformats the same value into base-10 scientific form so the magnitude is easier to read.