Geometric Sequence Calculator
Find the nth term and sum of a geometric sequence.
Results update as you type.
How it works
A geometric sequence is a list of numbers in which each term is found by multiplying the previous term by a constant called the common ratio. Starting from a first term of 2 with a ratio of 3, the sequence runs 2, 6, 18, 54, and so on, growing by a fixed factor at every step. This geometric sequence calculator returns the value of the term at any position n and the sum of all terms up to that position. Geometric progressions describe multiplicative growth and decay that appear everywhere: compound interest multiplies a balance by the same factor each period, populations and bacteria multiply, radioactive material halves repeatedly, and the resale value of equipment depreciates by a steady percentage. When the ratio is between minus one and one, the terms shrink toward zero and even an infinite sum settles on a finite value, which is the basis of many results in calculus and finance. Enter the first term, the common ratio, and how many terms you want, and the calculator gives both the nth term and the running total instantly, saving you from raising the ratio to a high power by hand.
aₙ = a·r^(n−1); Sₙ = a·(1 − rⁿ)/(1 − r)
In a geometric sequence each term equals the previous one multiplied by a fixed common ratio, so the nth term is the first term multiplied by the ratio raised to the power of n minus one. The sum of the first n terms uses the closed-form formula first term times one minus the ratio to the nth power, divided by one minus the ratio, which telescopes the repeated additions into a single expression; when the ratio equals one the sum is simply the first term times n.
Source:Geometric progression formulas
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Worked example
How to use this calculator
- 1
Enter your values
Enter first term (a), common ratio (r) and position (n).
- 2
Adjust the options
Fine-tune any options to match your situation.
- 3
Read your result
Nth term updates instantly as you type — no button to press.
Frequently asked questions
What is a geometric sequence?
A geometric sequence is an ordered list of numbers where each term after the first is obtained by multiplying the previous term by a fixed number called the common ratio. If the first term is 3 and the ratio is 2, the sequence is 3, 6, 12, 24, and so on. Unlike an arithmetic sequence, which adds a constant, a geometric one multiplies by a constant.
How do I find the nth term?
Multiply the first term by the common ratio raised to the power of one less than the position you want. For the fifth term of a sequence starting at 2 with ratio 3, you compute 2 times 3 to the fourth power, which is 2 times 81, or 162. The calculator does this exponentiation for you and returns the exact term.
How is the sum of the sequence calculated?
The sum of the first n terms uses a closed-form formula: the first term times the quantity one minus the ratio raised to the nth power, all divided by one minus the ratio. This avoids adding every term individually. When the common ratio is exactly one, every term is identical, so the sum simplifies to the first term multiplied by the number of terms.
What is the common ratio?
The common ratio is the fixed multiplier that turns each term into the next one. You find it by dividing any term by the term immediately before it. A ratio greater than one produces growth, a ratio between zero and one produces decay toward zero, and a negative ratio makes the terms alternate in sign while their magnitudes follow the same pattern.
Where are geometric sequences used?
They model anything that grows or shrinks by a constant percentage each period. Compound interest, loan balances, population growth, radioactive decay, depreciation of assets, and the reduction of drug concentration in the bloodstream all follow geometric patterns. Understanding the nth term and the sum lets you project future values and totals without stepping through every intermediate period by hand.