Math & Statistics
Arithmetic Sequence Calculator
Calculate the term at the entered count, any selected indexed term, the exact partial sum, the sequence mean, total change, and a comparison gap, with a term-by-term table for audit.
Term at entered count-
Term at selected index-
Sum through entered term count-
Mean of first and nth terms-
Nth term minus first term-
Partial sum minus comparison-
Nth term minus entered reference-
Selected term minus entered reference-
Decision view
Arithmetic sequence term and cumulative structure
Arithmetic sequence term and cumulative structureEqual additive steps build the indexed terms while the same blocks accumulate into the partial sum.
Exact scenario comparisonCommon difference changes while all other entered assumptions remain constant.
| Common difference | Term at entered count | Term at selected index | Sum through entered term count | Mean of first and nth terms | Nth term minus first term | Partial sum minus comparison | Nth term minus entered reference | Selected term minus entered reference |
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Period-by-period detail
Complete arithmetic-sequence table
How to use Arithmetic Sequence Calculator
- Enter the first term, signed common difference, and a positive whole number of terms.
- Choose a selected index within the modeled sequence when comparing one term with the total.
- Verify the closed-form sum against the final cumulative value in the detailed table.
Calculator guide
Understanding Arithmetic Sequence Calculator
An arithmetic sequence changes by one constant difference at every step, which makes each term, the selected term, and the cumulative sum independently verifiable.
Calculation method
How the calculation works
Apply a constant additive difference to generate any indexed term, use the closed-form arithmetic-series sum across the exact entered count, and compare both the nth and selected terms with the entered term-value reference. In the Arithmetic Sequence Calculator, the live scenario varies common difference and tracks term at entered count while the remaining results preserve the reconciliation path. Use a_n = a_1 + (n - 1)d for each term and S_n = n(a_1 + a_n)/2 for the sum of the first n terms.
Exact verification
Check the sequence with three invariants
A correct result can be tested without recomputing every term from scratch.
Worked situations
Practical examples
- Starting at 12 with a difference of 3.5 produces 15.5, 19, and 22.5 as the next three terms.
- A negative common difference creates a descending sequence without changing the formulas.
- When the first and final terms sum to zero, the sequence mean and total sum can be zero.
Better inputs
Useful tips
- Keep the index one-based: the first term is term 1.
- Retain exact decimals until the final display step.
- Use the cumulative table to locate where a budget, distance, or production target is crossed.
Before relying on the result
Limitations and common mistakes
- The model requires a constant additive difference and does not describe geometric or irregular sequences.
- Term count and selected index should be whole numbers.
- Selected share of sum is not meaningful when the partial sum is zero or near zero.
Reference
Key terms
- First term
- Value at index one.
- Common difference
- Constant amount added from one term to the next.
- Partial sum
- Total of the first entered number of terms.
- Selected index
- One-based position used for the selected-term result.
Important note
Calculated directly from the entered values using the displayed formula and rounding settings.
Frequently asked questions
Can the common difference be zero?
Yes; every term is then equal.
Can terms be negative?
Yes.
Why is the selected term outside the displayed range?
The selected index may exceed the entered term count; use an index within the modeled range for direct comparison.
Is an arithmetic sequence the same as linear growth?
For equally spaced integer indexes, its term values lie on a straight line.