Geometric Sequence Calculator

1. Choose First term & common ratio or Two known terms.
2. Enter the displayed values and Which term to solve for (n).
3. Click Calculate.
4. Read Result, Formula & steps, and First terms of the sequence.
5. Use Copy result or Download result (.txt) for the plain-text output.

Calculate a term and finite sum

Choose First term & common ratio, then enter First term (a₁), Common ratio (r), and Which term to solve for (n). The calculator applies a_n = a₁ * r^(n - 1). It also calculates the first n terms with S_n = a₁ * (1 - r^n) / (1 - r); when r = 1, it uses S_n = a₁ * n.

First term (a₁): 3
Common ratio (r): 2
Which term to solve for (n): 5

a₅ = 48
S₅ = 93

When the infinite sum appears

The finite sum is returned for every valid ratio. An Infinite sum appears only when |r| < 1, using S∞ = a₁ / (1 - r). Negative ratios and decimal values work when they are nonzero. The calculator treats values as plain numbers, so it does not attach, convert, or infer units.

Use two known terms

Choose Two known terms and enter Position of first known term (m), Value of that term (a_m), Position of second known term (k), and Value of that term (a_k). The positions must be different positive whole numbers, and the known values cannot be zero. The calculator derives r from a_m = a_k * r^(m - k), then works back to a₁.

When an even position gap and positive value ratio allow both signs, the calculation selects the positive root. It reports an error when the values require an even root of a negative number. The calculation uses the two terms you enter and assumes one constant ratio.

Input and display limits

• Value fields accept signed decimal text such as -2 and 0.5. Scientific notation, fractions, commas, and a leading plus sign are rejected.
• Positions and Which term to solve for (n) must be whole numbers from 1 through 1,000,000,000.
• First term (a₁) and Common ratio (r) cannot be zero.
• The First terms of the sequence section shows at most 50 terms, with an omission message when n is larger.
• Large powers or sums can display Infinity or NaN when the JavaScript number range is exceeded.

Changing the selected input mode hides the result. Reset clears the fields and returns to First term & common ratio. The calculation runs in your browser, and the script sends no entered values to a server.

Check the returned values

For First term (a₁) = 3, Common ratio (r) = 2, and Which term to solve for (n) = 5, the result includes aâ‚… = 48 and Sâ‚… = 93. The steps show the substitutions used for both values.

In Two known terms mode, Position of first known term (m) = 2 with Value of that term (a_m) = 6, and Position of second known term (k) = 5 with Value of that term (a_k) = 48, gives the ratio before the target term is calculated. Copy result keeps the displayed steps in one text value. Download result (.txt) saves that value as a text file.

When to use another math calculator

Browse Math Calculators for a different kind of calculation. Use the Arithmetic Sequence Calculator when terms change by a common difference. Choose the Sequence Calculator for arithmetic and geometric modes, missing values from supplied terms, or a broader sequence workflow.

Frequently asked questions