Geometric Progression Calculator

Find the n-th Term

Geometric Progression Formula

Formula

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

Where:

\(a_n\) — the n-th term of the progression
\(a_1\) — the first term
\(r\) — the common ratio (the constant each term is multiplied by)
\(n\) — the term number (a positive integer)

This formula gives the value of any term in a geometric progression directly, without computing all the terms before it. Each term equals the first term multiplied by the common ratio raised to the power \(n-1\).

Geometric Progression – Calculation Example

Example: a geometric progression starts at \(a_1 = 2\) and each term is 3 times the previous one (\(r = 3\)). Find the 5th term.

\[ a_5 = 2 \cdot 3^{5-1} = 2 \cdot 3^4 = 2 \cdot 81 = 162 \]

Result: \(a_5 = 162\). The full sequence is 2, 6, 18, 54, 162, …

How to Use the Geometric Progression Calculator

Enter the First term (a₁) — the value the sequence starts with, for example 2.
Enter the Common ratio (r) — the constant multiplier between consecutive terms, for example 3.
Enter the Term number (n) — which term you want to find; it must be a positive integer (1 or greater).
Read the result: the nᵗʰ term (aₙ) is calculated automatically as \(a_1 \cdot r^{n-1}\).

The result is rounded to three decimal places, which is enough for most classroom and practical problems. If you enter a term number smaller than 1, the calculator reports “Invalid n”.

What Is a Geometric Progression?

A geometric progression (also called a geometric sequence) is a sequence of numbers in which each term after the first is obtained by multiplying the previous term by a fixed non-zero constant called the common ratio \(r\). For example, with \(a_1 = 2\) and \(r = 3\), the sequence is 2, 6, 18, 54, 162, … The same ratio 3 appears between every pair of consecutive terms: \(6/2 = 18/6 = 54/18 = 3\).

The key property of a geometric progression is exponential behavior. If \(|r| > 1\), the terms grow (or grow in magnitude) very quickly; if \(0 < |r| < 1\), the terms shrink toward zero; if \(r = 1\), every term equals the first term; and if \(r\) is negative, the terms alternate in sign. This makes geometric progressions the natural model for anything that changes by a fixed percentage or factor per step — compound interest, population growth, radioactive decay, or repeatedly halving a quantity.

The formula \(a_n = a_1 \cdot r^{n-1}\) lets you jump straight to any term without listing the earlier ones. That is exactly what this calculator does: finding the 50th term of a progression takes the same effort as finding the 5th. If you also need the sum of the first \(n\) terms, it is given separately by \(S_n = a_1 \frac{r^n - 1}{r - 1}\) (for \(r \neq 1\)), which you can evaluate with the same inputs.

Interpreting the Result

The result \(a_n\) is simply the value of the term at position \(n\) in the sequence. Keep in mind how the inputs interact:

A larger \(|r|\) makes the terms grow or shrink faster; even small changes in \(r\) have large effects on distant terms.
A negative \(r\) produces alternating signs, e.g. \(a_1 = 2\), \(r = -3\) gives 2, −6, 18, −54, …
Fractional ratios such as \(r = 0.5\) model decay, e.g. 8, 4, 2, 1, 0.5, …

Always sanity-check the result: for \(n = 1\) the answer must equal \(a_1\), and for \(n = 2\) it must equal \(a_1 \cdot r\).

Assumptions, Limitations and Tips

n must be a positive integer. The calculator works with term positions 1, 2, 3, … and rejects values below 1.
r can be any real number — positive, negative, fractional, or 1. A ratio of exactly 1 makes every term identical.
Rounding: the answer is rounded to three decimal places. With very large \(n\) or large \(|r|\), results are shown in decimal notation and may be enormous.
This tool finds a single term. To sum the whole progression, apply the sum formula \(S_n = a_1 \frac{r^n - 1}{r - 1}\) with the same inputs.
Check for typos in r. Because the ratio is raised to a power, entering 0.3 instead of 3 changes the answer dramatically for large \(n\).

Geometric progressions often appear alongside other algebra topics. If you are working through series and sequences, you may also find our Fibonacci series calculator useful, or our tools for the proportion calculator and binomial coefficient when studying related counting and ratio problems.

Frequently Asked Questions

How do I find the common ratio if I only know some terms of the sequence?

Divide any term by the term before it: \(r = a_{k+1}/a_k\). For example, if the sequence contains 5 and 15 as consecutive terms, then \(r = 15/5 = 3\). Check the division on at least one more pair of terms to confirm the ratio is constant — that constancy is what makes the sequence geometric.

What is the difference between a geometric and an arithmetic progression?

In an arithmetic progression, each term is obtained by adding a fixed number (the common difference \(d\)): 2, 5, 8, 11, … In a geometric progression, each term is obtained by multiplying by a fixed ratio \(r\): 2, 6, 18, 54, … Arithmetic growth is linear; geometric growth is exponential. The formulas also differ: \(a_n = a_1 + (n-1)d\) versus \(a_n = a_1 \cdot r^{n-1}\).

What happens if the common ratio is negative or between 0 and 1?

A negative ratio makes the signs alternate: with \(a_1 = 2\) and \(r = -3\), the sequence is 2, −6, 18, −54, … A ratio with \(|r| < 1\), such as 0.5, shrinks the terms toward zero: 8, 4, 2, 1, 0.5, … This models decay processes like a medicine leaving the bloodstream or a car losing a fixed percentage of its value each year.

Can the first term or the common ratio be zero?

The first term can be zero, but then every term is zero regardless of \(r\), which is usually a sign of a mistyped input. A common ratio of zero is normally excluded from the definition of a geometric progression — it collapses the sequence to \(a_1, 0, 0, 0, \ldots\). Use a non-zero ratio; fractional values like 0.5 are perfectly valid.

How do I calculate the sum of the first n terms by hand?

Use \(S_n = a_1 \frac{r^n - 1}{r - 1}\) when \(r \neq 1\). Example: \(a_1 = 2\), \(r = 3\), \(n = 5\) gives \(S_5 = 2 \cdot \frac{3^5 - 1}{3 - 1} = 2 \cdot \frac{242}{2} = 242\). If \(r = 1\), the sum is simply \(S_n = n \cdot a_1\) because all terms are equal.

How do I verify my answer without redoing the whole calculation?

Two quick checks catch most errors. First, set \(n = 1\): the result must equal \(a_1\). Second, set \(n = 2\): the result must equal \(a_1 \cdot r\). You can also confirm that the term before your answer, divided into it, gives exactly \(r\). If these checks fail, the most likely cause is a typo in the ratio, since \(r\) is raised to a power and small input errors compound quickly.

Why do the terms get so huge when n is large?

The ratio is raised to the power \(n-1\), so the growth is exponential, not linear. With \(r = 3\) and \(n = 30\), you compute \(3^{29}\), which is already over 68 trillion. This is the same behavior seen in compound interest and population growth — doubling every step means a modest-looking sequence becomes astronomically large within a few dozen terms. Results are rounded to three decimal places for readability.