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Automatically calculate and generate a complete geometric sequence based on the first term, common ratio, and number of terms.
Overview
Understand what the tool solves, how it works, and the boundaries of its data.
A geometric sequence starts with a first term a₁ and gets each next term by multiplying the previous one by the same common ratio r. Enter those two values and a term count n to see the first n terms, the last term aₙ, and their total Sₙ. The result is a finite list and finite sum; it does not calculate an infinite series or infer a ratio from numbers you already have.
For example, starting at 3 and multiplying by 2 for five terms gives 3, 6, 12, 24, 48. The fifth term is 48 and the five terms add to 93. The tool updates these three results from the entered values and provides a copy control for each result.
The first term has not yet been multiplied by the ratio. Moving from term one to term n takes n−1 multiplications, so the general term is aₙ = a₁ × r^(n−1). That exponent is useful for checking the final item in the displayed sequence: if a₁=3, r=2, and n=5, then a₅=3×2⁴=48.
aₖ = a₁ × r^(k−1)
Sₙ = a₁ + a₁r + a₁r² + … + a₁r^(n−1)
For r ≠ 1: Sₙ = a₁(1−rⁿ)/(1−r). For r = 1: Sₙ = n×a₁.
The sum shown is the sum of the generated terms only. The formulas and definition are consistent with OpenStax’s treatment of geometric sequences and finite geometric sums; review the geometric sequence and series formulas (checked September 28, 2026).
If r=1, every term stays equal to the first term, so the sum is n×a₁. If r=0, the first term remains and every later term becomes zero. A negative ratio changes the sign at every multiplication: a₁=2, r=−3, and n=5 produce 2, −6, 18, −54, 162, with a sum of 122. These are direct consequences of repeated multiplication, not separate sequence modes.
Divide each term by the one before it and see whether the same ratio repeats. This check requires a nonzero preceding term. If the ratios differ, one common ratio does not describe the whole list.
No. The count represents how many discrete terms to generate, so use a positive whole number from 1 to 100.
Guide
Follow the workflow and verify inputs and outputs with practical examples.
Enter the starting value, such as 3. Zero, negative values, and decimals can be used as numeric inputs.
Enter the multiplier applied at every step, such as 2, 0.5, or −3. A common ratio is a multiplier; it is different from the fixed amount added in an arithmetic sequence.
Enter a positive whole-number count from 1 through 100. The result area recalculates as the inputs change; no separate generate action is needed.
Check aₙ against the last value in the comma-separated preview, and check Sₙ against the total you expect. Copy the individual value you need or copy the sequence for a worksheet or spreadsheet.
Use cases
See how the tool fits into real work and everyday tasks.
When a problem supplies a first term, ratio, and term count, compare the preview with your written terms, then use the final term and sum to catch an exponent or addition error.
For a lesson or study note, try a positive ratio for steady growth, a ratio between zero and one for shrinking magnitudes, or a negative ratio for alternating signs. Copy the list with the matching endpoint and total.
When a spreadsheet or script needs a sequence that changes by a fixed multiplier, generate up to 100 terms and paste the comma-separated values. Add units or dates separately if your task requires them.
Notes
Review scope, result limitations, and important precautions before use.
The sequence preview contains at most 100 terms. Very large ratios or long sequences can produce large values, scientific notation, or values beyond ordinary floating-point precision. Repeated decimal multiplication and addition can also create small rounding differences. If a result is used in a high-precision calculation, verify it with the precision rules required by that work.
The input fields use numbers, while the term count is read as an integer. A sequence with a zero first term or zero ratio can be generated by the recurrence rule, even though dividing adjacent terms to recover a ratio may be undefined. The output is a mathematical list only; it does not attach meaning, units, or dates to the values.
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