Geometric Series Calculator

Free Online Geometric Series Calculator

This geometric series calculator finds the sum of a finite geometric series (Sₙ) or an infinite geometric series (S∞) the instant you enter your values, and it can also work backward to find the first term, the common ratio, or the number of terms if you already know the total. Pick what you're solving for, type in the values you know, and the calculator rearranges the geometric series formula, draws a labeled diagram, and shows the complete step-by-step solution behind the answer.

A geometric series is the sum of the terms of a geometric sequence — a list of numbers where each term is found by multiplying the one before it by a fixed common ratio, r. For example, the sequence 2, 6, 18, 54, 162, 486 has a common ratio of 3, and its geometric series is the running total 2 + 6 + 18 + 54 + 162 + 486 = 728. This calculator handles that addition instantly, whether r is a whole number, a fraction, negative, or leads to an infinite series that converges to a finite limit.

Geometric Series Formula

Every calculation on this page traces back to two closely related core formulas — one for a finite number of terms, and one for an infinite series that converges:

• Finite sum formula: Sₙ = a₁(1 − rⁿ) / (1 − r), or Sₙ = a₁ × n when r = 1
• Infinite sum formula (only valid when |r| < 1): S∞ = a₁ / (1 − r)
• Solving for the first term (finite): a₁ = Sₙ(1 − r) / (1 − rⁿ)
• Solving for the first term (infinite): a₁ = S∞(1 − r)
• Solving for the common ratio (infinite): r = 1 − a₁/S∞
• Solving for the number of terms (finite, r > 0, r ≠ 1): n = ln(1 − Sₙ(1 − r)/a₁) / ln(r)

How to Use This Geometric Series Calculator

Start by picking what you want to solve for — the sum, the first term, the common ratio, or the number of terms — from the dropdown at the top. Except when solving for the number of terms, you can also choose whether you're working with a finite series (a fixed number of terms, n) or an infinite series (adding forever, valid only when |r| < 1). The input fields automatically relabel to match your choices, so you always know exactly which values to enter.

As soon as you type in your known values, every result updates instantly — the first term, common ratio, number of terms, last term, and the sum all appear in the results panel. For a finite series, the diagram below draws a log-scaled waterfall showing the running total build up column by column, with each bar labeled with the exact term added at that step. For an infinite series, the diagram instead shows the partial sums climbing toward a dashed line marking S∞, so you can visually see the series converge. Underneath, the step-by-step solution walks through exactly how the answer was derived from your entered numbers.

Worked Example: Finding the Sum of a Finite Series

Suppose a₁ = 2, the common ratio r = 3, and you want the sum of the first 6 terms. Using Sₙ = a₁(1 − rⁿ)/(1 − r), that's Sₙ = 2(1 − 3⁶)/(1 − 3) = 2(1 − 729)/(−2) = 2(−728)/(−2) = 728.

So the series 2 + 6 + 18 + 54 + 162 + 486 adds up to exactly 728 — the same numbers this calculator's default example uses, and the same total you can watch build up column by column in the waterfall diagram above.

Worked Example: Finding the Sum of an Infinite Series

Now suppose a₁ = 8 and the common ratio r = 0.5. Since |0.5| < 1, this infinite series converges, so S∞ = a₁/(1 − r) = 8/(1 − 0.5) = 8/0.5 = 16.

That means the series 8 + 4 + 2 + 1 + 0.5 + 0.25 + ... keeps adding smaller and smaller pieces forever, but the running total never quite reaches — and gets arbitrarily close to — 16. Switching this calculator's series length to Infinite and entering a₁ = 8, r = 0.5 reproduces this exact limit, and the convergence diagram shows the partial sums visibly closing in on that dashed S∞ line.

Worked Example: Finding the First Term from the Sum

Suppose you know a finite series has common ratio r = 2, runs for n = 5 terms, and sums to Sₙ = 93, but you don't know where it started. Rearranging gives a₁ = Sₙ(1 − r)/(1 − rⁿ) = 93(1 − 2)/(1 − 2⁵) = 93(−1)/(1 − 32) = −93/−31 = 3.

So the series begins at a₁ = 3 and doubles each time: 3, 6, 12, 24, 48 — and adding all five terms does total 93. Choosing 'First Term' from the solve-for dropdown and entering r = 2, n = 5, and Sₙ = 93 reproduces this exact result along with the full working.

Worked Example: Finding the Common Ratio

Suppose a series starts at a₁ = 5, runs for n = 4 terms, and sums to Sₙ = 200, but the common ratio is unknown. Because there's no simple algebraic shortcut for the ratio beyond two terms, this calculator scans a wide range of possible ratios and homes in on the exact value where the resulting series total matches your target — in this case, r = 3, since 5 + 15 + 45 + 135 = 200.

For an infinite series, though, solving for the ratio does have a clean formula: r = 1 − a₁/S∞. If a₁ = 4 and S∞ = 10, then r = 1 − 4/10 = 0.6, and indeed 4/(1 − 0.6) = 4/0.4 = 10 confirms the answer.

Worked Example: Finding the Number of Terms

Finally, suppose a₁ = 3, the common ratio r = 2, and the series sums to Sₙ = 381, but you don't know how many terms were added. Rearranging gives n = ln(1 − Sₙ(1 − r)/a₁) / ln(r) = ln(1 − 381(1 − 2)/3) / ln(2) = ln(1 + 127) / ln(2) = ln(128) / ln(2) = 7.

So it took exactly 7 terms — 3, 6, 12, 24, 48, 96, 192 — to reach a total of 381. This formula only works cleanly for a positive ratio not equal to 1; the calculator automatically checks these conditions and flags an error if the target sum doesn't land on a whole-number number of terms.

Understanding the Diagrams

The finite-series waterfall diagram uses a log-compressed vertical scale, because geometric growth can make later terms enormous compared to earlier ones — without compression, small early terms would be invisible next to a huge final term. Every column still shows its own exact, un-rounded value directly on the chart: the grey base is the total carried forward from every earlier term, and the colored cap — amber for the first term, indigo for the middle terms, green for the last — is the size of the new term just added, labeled as '+value'. A bold 'Σ=' label beneath each column shows the running total at that point.

The infinite-series convergence diagram instead plots ten successive partial sums as bars climbing toward a dashed green line marking S∞. Because |r| < 1 guarantees each new term is smaller than the last, the bars visibly get closer and closer to the dashed line without ever crossing it — a direct visual proof that the series converges rather than grows forever.

Finite vs. Infinite Geometric Series

A finite geometric series adds up a specific number of terms, n, and always produces a real, computable total regardless of the value of r. An infinite geometric series keeps adding terms forever, and it only has a finite sum — meaning it converges — when the common ratio's magnitude is strictly less than 1 (|r| < 1). If |r| ≥ 1, the terms don't shrink toward zero, so the running total either grows without bound or oscillates forever, and no finite S∞ exists.

This is why the 'infinite' option on this calculator is disabled from producing a result whenever |r| ≥ 1 — mathematically, there simply isn't a finite answer to give in that case, and the calculator says so directly instead of returning a misleading number.

Geometric Series vs. Geometric Sequence

A sequence is the ordered list of individual terms (2, 6, 18, 54), while a series is the sum of those terms added together (2 + 6 + 18 + 54 = 80). If you only need an individual term — like the 10th number in the pattern — rather than a running total, use the dedicated Geometric Sequence Calculator instead; this page is built specifically around the sum.

It's also easy to confuse geometric series with arithmetic series: in a geometric series, consecutive terms differ by a constant ratio (multiplication), while in an arithmetic series, consecutive terms differ by a constant amount (addition). If your series grows by adding the same amount each time rather than multiplying, use the Arithmetic Series Calculator instead.

Real-World Applications of Geometric Series

Geometric series show up constantly anywhere a quantity compounds by a fixed percentage or ratio, rather than growing by a fixed amount:

• Compound interest and investment growth, where a balance is repeatedly multiplied by (1 + interest rate) each period.
• Loan amortization and the present value of a series of future payments, both of which rely on finite geometric series formulas.
• Population or bacterial growth models where a population multiplies by a fixed growth factor each generation.
• Radioactive decay and half-life calculations, where a quantity is repeatedly multiplied by a decay factor less than 1.
• The 'bouncing ball' problem in physics, where each bounce reaches a fixed fraction of the previous height — an infinite geometric series gives the total distance traveled.
• Perpetuities and infinite payment streams in finance, which use the infinite geometric series formula directly to calculate present value.

Tips for Working with Geometric Series

Always check the common ratio's sign and magnitude before choosing a formula: a negative ratio makes the series alternate between positive and negative terms, while |r| < 1 versus |r| ≥ 1 determines whether an infinite version of the series even has a finite sum. When r = 1, the standard formula divides by zero, so the sum simplifies to the much simpler Sₙ = a₁ × n instead — this calculator switches to that formula automatically whenever r equals 1.

When solving for the common ratio from a target sum with more than two terms, remember there's no simple algebraic shortcut — the relationship becomes a higher-degree polynomial equation, which is why this calculator searches numerically rather than applying a single formula. If you already know the series is infinite, though, solving for r becomes a single clean step: r = 1 − a₁/S∞.