Geometric Sequence Calculator

Use this Geometric Sequence Calculator to enter values, adjust options, and review results in a compact responsive workspace.

Results are calculated automatically as you enter data.

First term a1 Common ratio r Term number n Sum through n

Formula and steps

nth term

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Assumptions and limitations

This Geometric Sequence Calculator is for general educational use. Results are estimates or mathematical outputs based on the inputs and stated assumptions, not professional engineering, safety, financial, legal, or academic advice. Check units, rounding, formulas, and limitations before using results for important work. For professional, safety-sensitive, construction, engineering, official, or high-cost uses, verify results with an appropriate method, current standards, required tolerances, and a qualified professional. Use of this calculator is subject to our Terms of Use.

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Related Calculators

What Is a Geometric Sequence?

A geometric sequence is a list of numbers where each term is found by multiplying the previous term by the same fixed value. That fixed multiplier is called the common ratio.

For example, this is a geometric sequence:

$$ 3,\ 6,\ 12,\ 24,\ 48,\ldots $$

Each term is multiplied by $2$ to get the next term, so the common ratio is $2$.

Geometric sequences are useful because they describe repeated multiplication. That makes them different from arithmetic sequences, which use repeated addition. Geometric patterns appear in exponential growth, exponential decay, repeated doubling, repeated halving, depreciation, scaling, population models, and many classroom sequence problems.

Why Geometric Sequences Matter

Geometric sequences help you describe patterns that change by a constant factor instead of a constant amount.

A geometric sequence can model situations such as:

a quantity doubling each step
a value being cut in half repeatedly
a population increasing by the same percentage each period
a balance, cost, or measurement being multiplied by the same factor over time
a pattern of terms that alternates signs because the ratio is negative

The main advantage is that you do not need to list every term. Once you know the first term and the common ratio, you can find a later term directly and add a finite number of starting terms with a formula.

Key Terms to Know

Sequence: An ordered list of numbers.
Term: One number in a sequence.
First term, $a_1$: The starting value of the sequence.
Common ratio, $r$: The multiplier used to move from one term to the next.
Nth term, $a_n$: The value of the term in position $n$.
Finite geometric series: The sum of a limited number of terms in a geometric sequence.
Partial sum, $S_n$: The sum of the first $n$ terms.
Growth: A pattern where the magnitude of terms increases because the ratio’s absolute value is greater than $1$.
Decay: A pattern where the magnitude of terms decreases because the ratio’s absolute value is between $0$ and $1$.

How Geometric Sequences and Finite Sums Work

A geometric sequence starts with $a_1$. To get each next term, multiply by the common ratio $r$.

The first few terms follow this pattern:

$$ a_1,\ a_1r,\ a_1r^2,\ a_1r^3,\ldots $$

The exponent is one less than the term number because the first term has not been multiplied by $r$ yet. The second term has been multiplied once, the third term twice, and so on.

The nth-term formula is:

$$ a_n = a_1r^{n-1} $$

Where:

$a_1$ is the first term.
$r$ is the common ratio.
$n$ is the term number.
$a_n$ is the value of the nth term.

A finite geometric sum adds the first several terms of the sequence. To avoid confusing the requested term number with the number of terms being added, let $m$ represent the number of terms in the sum:

$$ S_m = a_1 \cdot \frac{1-r^m}{1-r}, \quad r \ne 1 $$

Where:

$S_m$ is the sum of the first $m$ terms.
$m$ is the number of terms included in the sum.
$r \ne 1$ means the formula is used when the ratio is not $1$.

When $r = 1$, every term is the same as the first term. In that case, the finite sum is simpler:

$$ S_m = m \cdot a_1 $$

Examples of Geometric Sequences in Practice

Example 1: A Growth Sequence

Suppose the first term is $3$, the common ratio is $2$, and you want the 6th term.

The first terms are:

$$ 3,\ 6,\ 12,\ 24,\ 48,\ 96 $$

Use the nth-term formula:

$$ a_6 = 3 \cdot 2^{6-1} $$

$$ a_6 = 3 \cdot 2^5 = 96 $$

So the 6th term is $96$.

The sum of the first 6 terms is:

$$ S_6 = 3 \cdot \frac{1-2^6}{1-2} $$

$$ S_6 = 3 \cdot \frac{1-64}{-1} $$

$$ S_6 = 189 $$

This matches the direct addition:

$$ 3+6+12+24+48+96=189 $$

Example 2: A Decay Sequence

Suppose the first term is $64$, the common ratio is $0.5$, and you want the 7th term.

The first terms are:

$$ 64,\ 32,\ 16,\ 8,\ 4,\ 2,\ 1 $$

Use the nth-term formula:

$$ a_7 = 64 \cdot 0.5^{7-1} $$

$$ a_7 = 64 \cdot 0.5^6 = 1 $$

The sum of the first 7 terms is:

$$ S_7 = 64 \cdot \frac{1-0.5^7}{1-0.5} $$

$$ S_7 = 127 $$

This is a decay pattern because each term is half of the term before it.

Example 3: A Negative Ratio

Suppose the first term is $4$ and the common ratio is $-3$.

The first five terms are:

$$ 4,\ -12,\ 36,\ -108,\ 324 $$

The signs alternate because multiplying by a negative number changes the sign each time.

The 5th term is:

$$ a_5 = 4 \cdot (-3)^{5-1} $$

$$ a_5 = 4 \cdot (-3)^4 = 324 $$

The sum of the first 5 terms is:

$$ S_5 = 4 \cdot \frac{1-(-3)^5}{1-(-3)} $$

$$ S_5 = 4 \cdot \frac{244}{4} = 244 $$

A negative ratio does not mean the sequence is invalid. It means the terms switch between positive and negative values.

How to Interpret the Result

The nth-term result, $a_n$, is the value of one selected term. It does not include the terms before it.

The partial-sum result, $S_n$, is the total of a chosen number of starting terms. In the calculator, the term number used for $a_n$ and the number of terms used for the sum can be different, so read each result separately.

The first six terms and the preview graph show the beginning of the pattern. They are useful for seeing whether the sequence grows, decays, stays constant, collapses to zero, or alternates signs. They do not necessarily show the requested nth term if the term number is greater than $6$.

The common ratio controls the pattern:

If $r > 1$, the sequence grows in magnitude when the first term is positive.
If $0 < r < 1$, the sequence decays toward zero.
If $r = 1$, the sequence is constant.
If $r = 0$, the first term is followed by zeros.
If $r < 0$, the terms alternate signs.
If $|r| > 1$, the magnitudes grow.
If $0 < |r| < 1$, the magnitudes shrink.

If the first term is negative, the sign of the terms changes the visual pattern, but the ratio still determines the repeated multiplier.

Common Mistakes and Misconceptions

Confusing ratio with difference. A geometric sequence uses multiplication. An arithmetic sequence uses addition. The sequence $3, 6, 12, 24$ is geometric because each term is multiplied by $2$; the differences are not constant.

Using the wrong term number. The formula $a_n = a_1r^{n-1}$ uses $n-1$ in the exponent. A common mistake is using $r^n$, which shifts the result by one term.

Entering a decimal or zero for a term count. Term numbers and sum counts must be positive whole numbers. There is no 2.5th term in an ordinary sequence list, and this calculator starts counting at term $1$.

Typing a fraction expression instead of a decimal. If the ratio is one-half, enter $0.5$ rather than a fraction expression such as $1/2$.

Expecting the partial sum to be an infinite sum. The sum result is finite. It adds only the number of terms you choose.

Forgetting that negative ratios alternate signs. A ratio such as $-2$ multiplies the magnitude by $2$ and flips the sign each step.

Rounding too early. If a ratio is a long decimal, rounding it before calculating can change later terms noticeably, especially when the term number is large.

When to Use Geometric Sequences

Use geometric sequences when a pattern changes by repeated multiplication.

Common uses include:

finding a later term in a doubling, tripling, halving, or scaling pattern
studying exponential growth and decay in algebra
adding a finite number of terms in a geometric progression
checking whether a sequence has a constant ratio
comparing single-term values with cumulative totals
exploring alternating sign patterns from negative ratios
teaching or learning the difference between arithmetic and geometric sequences

Limitations and Things to Keep in Mind

This calculator assumes the sequence is real-valued and geometric. It does not test whether a random list of terms is geometric; it builds the sequence from the first term and common ratio you enter.

The calculator works with numeric inputs only. Symbolic expressions, exact algebraic forms, complex numbers, and fraction strings such as $1/2$ are not supported. Use decimals for fractional ratios.

Term numbers and sum counts must be positive whole numbers. Values such as $0$, negative term counts, and decimal term numbers are not valid for this calculator.

The sum is a finite partial sum. It does not calculate the limit of an infinite geometric series. For a nonzero first term, an infinite geometric series converges exactly when $|r|<1$, and its limiting sum is $a_1/(1-r)$; this calculator does not compute that limit.

Displayed results are rounded to a practical number of decimal places, so long decimal results should be treated as approximate. Very large ratios or very large term numbers can also produce values too large to display reliably.

The constant-sequence formula is used only when $r=1$ exactly. For ratios close to $1$, the calculator uses a numerically stable finite-sum calculation rather than treating a changing sequence as constant.

The preview graph shows only the first six terms. For larger term numbers, use the nth-term result and formula substitution rather than assuming the graph includes every term.

For schoolwork, finance-related modeling, engineering, safety, official records, or any high-stakes decision, double-check the inputs and formulas. A calculator can help with arithmetic, but it does not replace judgment about whether a geometric model is appropriate.

How to Use This Calculator

Enter the first term of the geometric sequence.
Enter the common ratio, which is the multiplier from one term to the next.
Enter the positive whole-number term $n$ whose value you want to find.
Enter the positive whole number of starting terms to include in the partial sum.
Review the nth-term result, the first six terms, and the partial-sum result.
Use the formula cards to see how your values fit into the nth-term and finite-sum formulas.
Use the growth or decay example presets to load sample values, clear the inputs when you want to start over, or download the preview graph as a PNG.

Frequently Asked Questions

What is the common ratio in a geometric sequence?

The common ratio is the fixed multiplier used to move from one term to the next. If a sequence goes $5, 15, 45, 135$, the common ratio is $3$ because each term is multiplied by $3$.

How is a geometric sequence different from an arithmetic sequence?

A geometric sequence changes by repeated multiplication. An arithmetic sequence changes by repeated addition. For example, $2, 4, 8, 16$ is geometric, while $2, 4, 6, 8$ is arithmetic.

Why does the nth-term formula use $n-1$?

The first term is already given as $a_1$, so it has been multiplied by the ratio zero times. The second term has one multiplication, the third term has two, and the nth term has $n-1$ multiplications.

What happens when the common ratio is $1$?

When $r=1$, every term equals the first term. The nth term is $a_1$, and the sum of the first $m$ terms is $m \cdot a_1$.

What happens when the common ratio is $0$?

When $r=0$, the first term stays as entered and every term after it becomes $0$. For example, if $a_1=9$ and $r=0$, the sequence is $9, 0, 0, 0,\ldots$.

Is the partial sum the same as an infinite geometric series?

No. The partial sum adds only a finite number of starting terms. For a nonzero first term, an infinite geometric series converges only when $|r|<1$, with limiting sum $a_1/(1-r)$. This calculator still reports finite partial sums only.

Sources and References

Books

Jay Abramson. College Algebra 2e. OpenStax, 2021. Chapter 9.3, “Geometric Sequences,” and Chapter 9.4, “Series and Their Notations.” Chapter 9.3 and Chapter 9.4.
Lynn Marecek and Andrea Honeycutt Mathis. Intermediate Algebra 2e. OpenStax, 2020. Chapter 12.3, “Geometric Sequences and Series.” Chapter 12.3.
Donna Kirk. Contemporary Mathematics. OpenStax, 2023. Section 3.11, “Geometric Sequences.” Section 3.11.