Geometric Sequence Calculator

Free Online Geometric Sequence Calculator

This geometric sequence calculator finds any missing piece of a geometric sequence — the nth term, the common ratio, the first term, or the total number of terms — the moment you enter the values you already know. It also calculates the sum of the first n terms automatically, draws a labeled growth diagram so you can see every term's exact value at a glance, and shows the complete step-by-step solution behind the answer.

Unlike a basic nth term calculator that only works in one fixed direction, this tool lets you pick exactly what you're solving for. Know the first term, common ratio, and how many terms you want? Solve for the nth term and sum. Know two terms and their positions but not the common ratio? Solve for r instead. Whatever equation you're working with, the calculator rearranges the geometric sequence formula for you and returns an exact, fully worked answer.

What Is a Geometric Sequence?

A geometric sequence (also called a geometric progression) is a list of numbers in which each term after the first is found by multiplying the previous term by a fixed, constant value called the common ratio, written as r. For example, 2, 6, 18, 54, 162 is a geometric sequence because each term is exactly 3 times the term before it — here, r = 3.

The common ratio can be greater than 1 (rapid growth), between 0 and 1 (steady decay toward zero), negative (the sequence alternates between positive and negative values), or even exactly 1 (a constant sequence). Recognizing the common ratio is the first step to solving almost any geometric sequence problem, and it's exactly what this calculator can find for you if you already know two terms in the sequence.

Geometric Sequence Formulas

Every geometric sequence calculation traces back to one core formula, which can be rearranged to solve for whichever variable is missing:

• nth term formula: aₙ = a₁ × r^(n − 1)
• Sum of n terms (r ≠ 1): Sₙ = a₁ × (1 − rⁿ) / (1 − r)
• Sum of n terms (r = 1): Sₙ = n × a₁
• Solving for the common ratio: r = (aₙ / a₁)^(1 / (n − 1))
• Solving for the first term: a₁ = aₙ / r^(n − 1)
• Solving for the number of terms: n = log(aₙ / a₁) / log(r) + 1

How to Use This Geometric Sequence Calculator

Start by choosing what you want to solve for from the dropdown at the top of the calculator: the nth term and sum, the common ratio, the first term, or the number of terms. The input fields automatically relabel and show only the values you need to enter for that mode — you never have to guess which formula to rearrange by hand.

Type in your known values, and every result updates instantly: the first term, common ratio, term position, the nth term, and the sum of the sequence all appear in the results panel. Below that, the growth diagram renders the sequence as a series of bars — each one labeled with its exact value — with a small '×r' arrow between every pair of bars showing the constant multiplier. Finally, the step-by-step solution spells out precisely how the answer was derived from your numbers, formula by formula.

Worked Example: Finding the nth Term and Sum

Suppose a1 = 2 and the common ratio r = 3, and you want to find the 6th term along with the sum of the first 6 terms. Using aₙ = a₁ × r^(n − 1), the 6th term is a₆ = 2 × 3^(6 − 1) = 2 × 243 = 486.

For the sum, Sₙ = a₁ × (1 − rⁿ) / (1 − r) = 2 × (1 − 3⁶) / (1 − 3) = 2 × (1 − 729) / (−2) = 2 × (−728) / (−2) = 728. So the sequence runs 2, 6, 18, 54, 162, 486, and the first six terms add up to exactly 728 — the same numbers this calculator's default example uses.

Worked Example: Finding the Common Ratio

Now suppose you know a1 = 5 and the 4th term a4 = 135, but you don't know the common ratio. Rearranging the formula gives r = (aₙ / a₁)^(1 / (n − 1)) = (135 / 5)^(1 / (4 − 1)) = 27^(1/3) = 3.

That means the sequence is 5, 15, 45, 135 — each term is exactly 3 times the one before it. Switching the calculator's 'Solve for' dropdown to Common Ratio and entering a1 = 5, a4 = 135, and n = 4 reproduces this exact result, along with the full working.

Worked Example: Finding the Number of Terms

Sometimes you know the first term and the common ratio, along with a target value, but you need to know which position that value occupies in the sequence. Suppose a1 = 4, the common ratio r = 2, and you want to know which term equals 512. Rearranging the formula gives n = log(aₙ / a₁) / log(r) + 1 = log(512 / 4) / log(2) + 1 = log(128) / log(2) + 1 = 7 + 1 = 8.

So 512 is the 8th term of the sequence 4, 8, 16, 32, 64, 128, 256, 512. If this calculation had produced a decimal instead of a whole number, that would mean 512 simply isn't a term of this particular sequence at all — the calculator automatically flags that case for you instead of returning a misleading fractional position.

Worked Example: Finding the First Term

Finally, suppose you're told the common ratio is r = 0.5, the 5th term is a5 = 3, and you need to find where the sequence started. Rearranging gives a1 = aₙ / r^(n − 1) = 3 / 0.5^(5 − 1) = 3 / 0.0625 = 48.

So the sequence begins at a1 = 48 and is halved each time: 48, 24, 12, 6, 3 — confirming that 3 is indeed the 5th term. This mode is especially useful in word problems that describe a sequence's ending value, such as a decaying quantity, but never state where it actually started.

Understanding the Growth Diagram

The diagram is designed so you can understand the sequence just by looking at it, without needing to read a single number in text. Each bar represents one term of the sequence, colored by role — amber for the first term (a₁), indigo for the middle terms, and green for the target nth term (aₙ) — and every bar is labeled directly with its own exact value, so nothing is hidden behind an abstract axis or scale.

Because geometric sequences can grow or shrink extremely fast, the bar heights use a compressed (logarithmic) scale internally so that both the smallest and largest terms stay visible on the same chart — but every printed number is still the true, exact value, never rounded to fit the scale. The pink '×r' arrows between consecutive bars show the common ratio repeated at every single step, visually proving that the sequence is multiplied by the exact same factor every time. For sequences with many terms, the diagram automatically compresses the middle into an ellipsis so the first few terms and the final target term (aₙ) stay clearly visible without the chart becoming cluttered.

Geometric vs. Arithmetic Sequences

It's easy to confuse geometric sequences with arithmetic sequences, but the difference is simple: in a geometric sequence, consecutive terms differ by a constant ratio (multiplication), while in an arithmetic sequence, consecutive terms differ by a constant amount (addition). For example, 2, 4, 8, 16 is geometric (multiply by 2 each time), while 2, 4, 6, 8 is arithmetic (add 2 each time).

If your sequence is growing or shrinking by adding a fixed amount rather than multiplying, use our dedicated Arithmetic Sequence Calculator instead, which applies the aₙ = a₁ + (n − 1)d formula rather than the multiplication-based formula used here.

Real-World Applications of Geometric Sequences

Geometric sequences show up constantly outside the math classroom, anywhere something grows or shrinks by a fixed percentage or multiplying factor rather than a fixed amount:

• Compound interest, where a savings or investment balance is multiplied by the same growth factor every compounding period.
• Population growth models, where a population multiplies by a roughly constant growth rate each generation or year.
• Radioactive decay and half-life problems, where a quantity is repeatedly multiplied by 0.5 (or another decay factor) over equal time intervals.
• The height of a bouncing ball, where each bounce reaches a fixed fraction of the height of the bounce before it.
• Viral or exponential spread, such as a video, rumor, or infection multiplying by a roughly constant factor with each new round.
• Depreciation of assets like vehicles or equipment, where the value is multiplied by the same retention factor every year.

Tips for Working with Geometric Sequences

Always double-check the sign and size of your common ratio — a ratio between −1 and 1 means the sequence shrinks toward zero, a ratio greater than 1 means rapid growth, and a negative ratio means the terms alternate between positive and negative. Mixing up these cases is the most common source of errors when solving by hand.

Remember that n always represents a whole-number position (1st term, 2nd term, and so on), so if a calculated value of n comes out as a decimal, it usually means the target value you entered isn't actually part of that sequence. Also note that if |r| < 1, the sum of infinitely many terms converges to a finite value instead of growing forever — for that calculation specifically, use our dedicated Geometric Series Calculator, which handles both finite and infinite geometric sums.