Free Geometric Sequence Calculator

What Is a Geometric Sequence?

A geometric sequence (also called a geometric progression) is an ordered list of numbers where, after the first, every term is the product of the previous term and a fixed, non‑zero value known as the common ratio ($r$). For instance, $3, 6, 12, 24, \ldots$ is geometric because each term is twice the previous one – here $r = 2$. The first term is often denoted $a_1$.

The two essential ingredients of any geometric progression are $a_1$ and $r$. With this free geometric progression calculator, you supply these (or any other pair of known values) and obtain the entire sequence, the sum of a finite or infinite number of terms, and more.

Explicit and Recursive Formulas

There are two standard ways to characterise a geometric sequence algebraically.

Explicit Formula

The explicit (or closed‑form) formula lets you compute any term directly:

$$a_n = a_1 \cdot r^{\,n-1},$$

where $n$ is a positive integer ($n = 1,2,3,\dots$). This is the most efficient way to find, say, the 100th term without listing the 99 preceding ones.

Recursive Formula

The recursive definition expresses each term using the previous term:

$$a_n = a_{n-1} \cdot r, \qquad a_1 \text{ given}.$$

The recursive form mirrors the natural generation of the sequence – multiply by $r$ repeatedly – but it requires the first term to start.

Both formulas describe the same progression; the choice depends on the context (e.g., programming loops often use recursion, while analytical tasks prefer the explicit form).

How to Work With the Calculator

Using this geometric series calculator is straightforward. You select the information you already have:

• $a_1$ and $r$,
• $r$ and any one term $a_n$, or
• two arbitrary terms of the sequence.

After entering the data, the calculator derives the missing parameters and displays the progression. You can control how many terms are shown and which range to display. The tool also functions as a sum of geometric sequence calculator: you specify the starting and ending indices for a partial sum, or request the infinite sum (when it exists).

Sum of a Finite Geometric Series

Adding the first $n$ terms of a geometric progression gives a finite geometric series. The sum is obtained with the formula:

$$S_n = a_1 \cdot \frac{1 - r^{\,n}}{1 - r}, \qquad r \neq 1.$$

If $r = 1$, the progression is constant and $S_n = n \cdot a_1$.

This formula can be derived by multiplying the series by $(1-r)$ and solving a simple equation – a classic trick that avoids adding each term manually.

Sum of an Infinite Geometric Series

An infinite geometric series converges (has a finite sum) only when the common ratio satisfies $|r| < 1$. In that case the sum is:

$$S_{\infty} = \frac{a_1}{1 - r}.$$

When $|r| \geq 1$, the terms either grow without bound or oscillate, so the series diverges. The calculator automatically checks the convergence and reports the infinite sum only when it exists.

Practical Examples

Zeno’s Dichotomy Paradox

Zeno’s ancient puzzle illustrates how an infinite geometric series can have a finite total. Suppose you walk from A to B in time $t$. The first half takes $t/2$, the next quarter takes $t/4$, and so on. The times form a geometric sequence with $a_1 = t/2$ and $r = 1/2$. Summing to infinity:

$$S_{\infty} = \frac{t/2}{1 - 1/2} = t.$$

Thus the infinite number of steps still amounts to a finite travel time.

Applications in Finance and Science

Geometric sequences appear in compound interest calculations (each period multiplies the balance by $1 + \text{rate}$), exponential population growth, radioactive decay, and many other fields. The free geometric sequence calculator can handle any such progression quickly.

The Alternating Series

If the common ratio is negative, the terms alternate sign. For example, $1, -1, 1, -1, \dots$ has $r = -1$. Its partial sums flip between 1 and 0, so the infinite series does not converge. The tool will indicate divergence for such cases.

Summary Table: Key Formulas

Quantity	Formula
$n$-th term (explicit)	$a_n = a_1 r^{\,n-1}$
Recursive definition	$a_n = a_{n-1} r$ ($a_1$ known)
Sum of first $n$ terms ($r \neq 1$)	$S_n = a_1 \dfrac{1 - r^n}{1 - r}$
Sum to infinity ($	r

Whether you need a geometric progression calculator to find the common ratio, a geometric series calculator to sum the terms, or a tool to explore the sum of geometric sequence for homework, this online resource provides all the answers in seconds.