Geometric Series Calculator

How a Geometric Series Is Built

This geometric series calculator works with a geometric sequence: every term after the first is the preceding term multiplied by one fixed common ratio. With first term $a_1$ and ratio $r$, the sequence is $a_1,a_{1}r,a_1{r}^2,...$. Its nth term is $a_n=a_1{r}^{n-1}$. For the first $n$ terms, the finite sum is $S_n=a_1\frac{1-r^n}{1-r}$ when $r\ne 1$. A ratio above one magnifies successive terms, while a ratio whose magnitude is below one makes them shrink.

The finite geometric-series formula comes from aligning the same sequence at two scales. Write $S_n=a_1+a_{1}r+a_1{r}^2+...+a_1{r}^{n-1}$. Multiplying by $r$ gives $r{S}_n=a_{1}r+a_1{r}^2+...+a_1{r}^n$. Subtraction cancels the middle terms: $S_n-r{S}_n=a_1-a_1{r}^n$. Thus $S_n(1-r)=a_1(1-r^n)$, which yields the displayed formula after division by $1-r$.

Geometric series describe repeated multiplication rather than repeated addition. Compound interest uses a fixed growth factor per compounding period; a fixed fractional loss can model a decaying quantity; and divide-and-conquer algorithms often create successively smaller blocks of work. The calculator is intended for the finite sequence specified by the three fields, so its result is a term and a partial sum, not automatically an infinite-series value.

The inputs on this page map directly to the calculation. Enter $a_1$, $r$, and a positive integer $n$; the result reports $a_n$ as $a_1{r}^{n-1}$ and the sum through that term. When the ratio is exactly one, every term equals the first term, $a_n=a_1$, and the JavaScript uses $a_{1}n$ for the sum. Otherwise it uses the finite-sum expression above.

An infinite geometric series is a separate question from the finite total calculated here. If , increasingly distant terms approach zero and the infinite sum is $S=\frac{a_1}{1-r}$. If $\left|r\right|\ge 1$, the ordinary infinite series does not converge. The term count entered in this calculator still defines a valid finite partial sum in either case.

For a concrete geometric-series calculation, let $a_1=5$ and $r=3$. The terms begin 5, 15, 45, and 135. The fourth term is $a_4=53^3=135$, while the sum of the first four terms is $S_4=5\frac{1-3^4}{1-3}=200$. This illustrates why a ratio greater than one can make both individual terms and partial sums rise quickly.

This geometric sequence table uses $a_1=2$ and $r=1.5$:

Here each term is 1.5 times the preceding term, so the sequence rises by 50% at every step. Changing only the ratio lets you compare growth, decay, and alternating signs. A negative ratio alternates the signs of successive terms; a ratio of zero makes every term after the first zero. These cases are useful checks when interpreting a finite total rather than relying only on its magnitude.

Geometric scaling also appears in digital graphics, cache hierarchies, and musical frequency ratios. In each setting, the key modeling decision is whether a single multiplier is genuinely constant from one step to the next. When it is not, a geometric-series formula can be a useful approximation but does not exactly describe the process.

Because this calculator uses JavaScript number arithmetic, very large exponents or extreme ratios can be affected by floating-point rounding, overflow, or underflow. For ordinary classroom calculations, the displayed four decimal places are often convenient; for precision-sensitive work, retain more precision independently and verify the assumptions behind the ratio. Check especially that the term count means the number of included terms, because the exponent on the reported nth term is one less than that count.

Worked Example: A Decaying Geometric Series

For a geometric series with first term 12, ratio 0.8, and eight terms, the eighth term is $12{0.8}^{8-1}$, approximately 2.46. The sum through eight terms is $S_8=12\frac{1-{0.8}^8}{1-0.8}$, approximately 49.66. The finite sum adds the eight terms; it is not the amount remaining after the eighth multiplication, which is represented by the eighth term. The recurrence $a_n=r{a}_{n-1}$ provides an additional way to check each step in this decaying sequence.

Geometric Series Compared with Arithmetic Series

A geometric series multiplies by a common ratio at each step, unlike an arithmetic series, which adds a constant difference. Starting both sequences at 5 makes the contrast clear:

The arithmetic values change by the same amount, whereas the geometric values are repeatedly scaled. This distinction determines whether a problem calls for a common difference or the common-ratio input used by this calculator.

Geometric Series in Practical Models

Geometric-series reasoning is useful whenever equal intervals apply the same multiplier: recurring interest factors, repeated percentage depreciation, echo amplitudes, fractal levels, and some branching models are familiar examples. The first term establishes the starting scale, the ratio controls the repeated change, and the number of terms specifies how far the finite model extends.

Limitations and Assumptions for Geometric Series

This geometric series calculator assumes one constant ratio and a finite, positive number of terms. Variable rates, fees, capacity limits, or outside additions require a different model. Small floating-point discrepancies are possible when $n$ is large or $r$ is extreme, so important results should be checked with the relevant domain assumptions and an appropriate precision method.

Related Calculators for Multiplicative Growth

Continue exploring repeated multiplication with the Compound Interest Calculator and the Exponential Growth and Decay Calculator.