Geometric Sequence
The nth term and sum of a geometric sequence.
8th term (aₙ)
384
sum of first 8 = 765
Sum of terms
765
Common ratio
2
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How the Geometric Sequence works
This calculator finds the nth term and the sum of a geometric sequence given a first term, common ratio, and number of terms — making it an essential tool for students, teachers, engineers, and anyone working with exponential growth or decay patterns.
A geometric sequence is a list of numbers where each term is obtained by multiplying the previous term by a fixed value called the common ratio. For example, the sequence 3, 6, 12, 24, 48 has a first term of 3 and a common ratio of 2. Unlike an arithmetic sequence — where you add a constant difference — a geometric sequence scales multiplicatively, which is why it models exponential phenomena so naturally. This calculator requires just three inputs: the first term (a), the common ratio (r), and the number of terms (n) you want to evaluate.
The nth term formula, a·r^(n−1), allows you to jump directly to any position in the sequence without listing every preceding term. This is particularly powerful when n is large — for instance, finding the 50th term of a sequence would be tedious by hand but is instantaneous here. The common ratio is the single most influential factor: a ratio greater than 1 produces exponential growth, a ratio between 0 and 1 produces exponential decay, a negative ratio causes the sequence to alternate between positive and negative values, and a ratio of exactly 1 collapses the sequence into a constant series.
The sum of the first n terms, known as a partial sum, uses the formula a·(1−r^n)/(1−r) when r ≠ 1. This formula elegantly condenses what could be hundreds of additions into a single computation. A common mistake is forgetting that this formula breaks down when r = 1 — in that special case, every term equals a, so the sum is simply n·a. Another frequent error is confusing the common ratio with the common difference from arithmetic sequences; always verify your ratio by dividing any term by the one before it, not subtracting.
For infinite geometric series — where n approaches infinity — convergence only occurs when the absolute value of r is less than 1. In that scenario, the sum converges to a/(1−r), a formula widely used in finance, physics, and probability. This calculator focuses on finite sequences, but understanding the convergence condition helps you recognize when a real-world model is stable versus one that grows without bound. Always sanity-check your ratio before computing: a ratio of −3 will produce wildly oscillating, rapidly growing terms, while a ratio of 0.5 will produce a series that halves with each step and converges quickly.
Formula
aₙ = a₁·r^(n−1) · Sₙ = a₁·(rⁿ − 1)/(r − 1)
Pro tips
- Always verify your common ratio by dividing at least two consecutive pairs of terms — if you get different values, the sequence is not geometric and these formulas will not apply.
- When working with financial problems like compound interest, recognize that the balance after each period forms a geometric sequence where r = (1 + interest rate), making this calculator directly applicable.
- Watch for a negative common ratio: the sequence will alternate in sign, and its partial sum can behave counter-intuitively — summing many terms may give a result smaller than the first term alone.
- For very large exponents, rounding errors can accumulate in manual calculations; this calculator handles the precision automatically, but always confirm your ratio is entered as an exact decimal (e.g., 0.5, not 1/2).
- If your result for the nth term is astronomically large or approaches zero faster than expected, double-check whether you intended exponential growth (r > 1) or decay (0 < r < 1) — swapping these is one of the most common input errors.
Key terms
- First Term (a)
- — The starting value of the geometric sequence, from which all subsequent terms are generated by repeated multiplication.
- Common Ratio (r)
- — The fixed multiplier applied to each term to produce the next; calculated by dividing any term by its preceding term.
- nth Term
- — The value at a specific position n in the sequence, given directly by the formula a · r^(n−1) without needing to list all prior terms.
- Partial Sum S(n)
- — The total obtained by adding together the first n terms of a geometric sequence, computed efficiently using the closed-form summation formula.
- Geometric Series
- — The expression formed when the terms of a geometric sequence are written as a sum rather than just a list.
- Convergence
- — The property of an infinite geometric series to approach a finite limiting value, which only occurs when the absolute value of the common ratio is strictly less than 1.
