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Enter your values, then click Calculate result.How this calculator helps
Calculate the finite sum and final term of an arithmetic or geometric sequence from a first term, step or ratio and term count. Enter the first included term, the difference or ratio, and an integer count from one through one thousand. The type selector determines how the second number is used. Signed and decimal terms are supported within the input bounds. The count describes how many terms are summed, including the first; it is not the number of gaps between terms or the index of a term in an unspecified earlier sequence.
How to use it
- 1
Select the correct input roles for sequence sum calculator and enter the values described below. The demonstration defaults illustrate the method; they are not independently verified personal measurements or live market data.
- 2
Review arithmetic progression and the original source record. Match units, signs and the chosen mode before submitting, rather than relying on a familiar-looking default number.
- 3
Click Calculate result to submit the current fields. Editing an input preserves the prior submitted output until you calculate again; the pending-change message distinguishes that saved output from the new values.
- 4
Check the labeled output against the worked example and independent verification steps. Review supported scope before using the result in another document, and keep the complete input basis with a copied answer.
Formula and methodology
Arithmetic: aₙ=a₁+(n−1)d; Sₙ=n(a₁+aₙ)/2. Geometric: aₙ=a₁rⁿ⁻¹; Sₙ=a₁(1−rⁿ)/(1−r), with Sₙ=na₁ when r=1.
The calculator applies the displayed arithmetic to the values entered on this device. It does not silently load a local tax rate, currency conversion or commercial assumption.
Worked calculation example
An arithmetic sequence starting at 3 with difference 2 and five terms is 3,5,7,9,11. Its sum is 35 and final term 11. A geometric sequence starting at 3 with ratio 2 and five terms is 3,6,12,24,48, with sum 93. The same written step value has an entirely different meaning under the two selectors.
How to interpret your result
Geometric terms multiply by the same ratio. A negative ratio alternates signs, zero makes every term after the first zero, and one keeps all terms equal. A ratio between zero and one reduces positive term magnitudes. This page sums only the chosen finite number of terms, even when an infinite-series limit exists. It does not interpret the count as infinity or require a convergent ratio.
For different inputs or formulas, use Lottery Annuity Calculator; Compound Interest Calculator — Monthly Contributions and ETF Scenarios; Time Value of Money Calculator.
Related questions this calculator covers
- summation calculator
- sum calculator
- arithmetic sequence calculator
Scenario comparison
| Scenario | What it shows |
|---|---|
| Constant | first term 4 with difference zero and five terms sums to 20. |
| Alternating | first term 1, ratio −1 and four terms sums to zero. |
| One term | any supported step leaves sum equal to the first term. |
Common mistakes to avoid
- Using n differences for n terms.
- Confusing additive difference with multiplicative ratio.
- Interpreting a finite geometric sum as an infinite limit.
Write a short sequence manually and add its terms. Verify the arithmetic sum with first-plus-last pairing and the geometric sum with its closed form when the ratio differs from one. Check ratios zero, one and minus one separately. The final term should match exactly n−1 step operations from the first, allowing floating-point rounding.
Authoritative reference. Method and scope reviewed on 5 October 2026. SolvePilot provides the original worked example and bounded browser implementation. Editorial and arithmetic review by Mohammad Qasim does not certify user measurements, a real contract or an individual professional decision. The reference supplies method, unit or source-record context; it does not approve this implementation or its inputs.What can affect the result?
Finite sequence inputs
Enter the first included term, the difference or ratio, and an integer count from one through one thousand. The type selector determines how the second number is used. Signed and decimal terms are supported within the input bounds. The count describes how many terms are summed, including the first; it is not the number of gaps between terms or the index of a term in an unspecified earlier sequence.
Arithmetic progression
Arithmetic terms increase by the same additive difference. A negative difference decreases the terms and can cross zero. The final term uses n−1 steps because the first term is already included. The sum can be checked by pairing the first and last terms, whose average is the sequence average. A zero difference creates a constant sequence with sum equal to first term multiplied by count.
Geometric progression
Geometric terms multiply by the same ratio. A negative ratio alternates signs, zero makes every term after the first zero, and one keeps all terms equal. A ratio between zero and one reduces positive term magnitudes. This page sums only the chosen finite number of terms, even when an infinite-series limit exists. It does not interpret the count as infinity or require a convergent ratio.
Numerical implementation
The worksheet iterates through the bounded term count and uses compensated summation to reduce avoidable rounding loss. That also handles the geometric ratio-one case without dividing by zero. An intermediate or total value outside finite browser precision is rejected. Cancellation between large positive and negative terms can still make relative precision poor, so a displayed decimal result is not an exact symbolic proof.
Supported scope
This tool covers complete arithmetic or geometric progressions with one starting term and fixed step rule. Arbitrary sigma expressions, variable ratios, omitted terms and recursive definitions need another method. The final-term row helps verify whether your sequence definition matches the inputs. Financial applications require a separate timing and currency model; a bare geometric sum does not automatically represent a present value or actual investment return.
Privacy and browser processing
Values entered on this page are processed in the current browser session. SolvePilot does not require an account and does not receive the values entered into the calculator. Refreshing or closing the page clears the working values unless the browser itself restores a previous session. Avoid entering identifying or account information because the calculation needs summary values only.
Accuracy and verification
Accuracy depends first on input quality. Confirm definitions, scales, dates and source information before entering a value. Keep an independent record of any result used for planning because this page does not create an official statement or retain a calculation history.
Limits of this estimate
This tool covers complete arithmetic or geometric progressions with one starting term and fixed step rule. Arbitrary sigma expressions, variable ratios, omitted terms and recursive definitions need another method. The final-term row helps verify whether your sequence definition matches the inputs. Financial applications require a separate timing and currency model; a bare geometric sum does not automatically represent a present value or actual investment return.
Sources and review information
Frequently asked questions
Does the count include the first term?＋
Yes. A count of one returns the first term as both sum and final term.
What if the geometric ratio is one?＋
Every term equals the first, so the sum is first term times count. The implementation handles that boundary directly.
Can a geometric ratio be negative?＋
Yes. Signs alternate according to the repeated multiplication, and the finite sum can be positive, negative or zero.
Does it sum an infinite series?＋
No. The count must be a finite integer from one through one thousand.
Can I enter a sigma expression?＋
No. Enter the first term, fixed difference or ratio, and count for one of the two supported progressions.