Free · Sequences
Arithmetic Sequence Sum and Terms
Find the sum and final value of an arithmetic sequence, with signed terms, constant differences and up to 200 listed terms.
- Formula & worked example
- Private in your browser
- No signup
Calculator inputs
Loading calculator…
How to use this calculator
- Enter the known values in the units shown. Results update as you type.
- Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
- Use the worked example to check the method. Reset restores the starting fields.
Private by default
Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.
Formula and method
Starting at1 with difference 1 for 5 terms gives final value5 and sum15.
aₙ=a₁+(n−1)d; Sₙ=n(2a₁+(n−1)d)/2.
Worked example
Enter these known values and leave the other values blank.
- Initial value
- 1
- Difference
- 1
- Periods
- 5
- Final value
- 5
- Sum
- 15
Assumptions and limitations
- The number of terms is a positive integer. Initial value, difference, final value and sum may be zero or negative. All five captured controls remain editable; clear an unknown before solving backwards.
- The term list is available for1 to 200 terms. Larger counts still have an analytic sum and final value; the list is omitted rather than truncated.
- Listed terms intentionally correct the source’s JavaScript Number coercion and use the same64 digits of decimal working precision as the analytic result. This preserves large integer units and nonzero tiny values, and avoids binary decimal artifacts. The list uses ordinary numeric text with scientific notation when needed.
- Controls and result summaries follow the captured relative display rounding. The Calculated line and listed terms retain more of the computed value; above 200 terms the Calculated line still provides the scalar value.
- Count inversion can have two positive integer answers. The captured plus-square-root branch is retained when valid; if only the other root is admissible it is used. A displayed count is not a uniqueness claim.
- For difference 0 and nonzero initial value, a known sum determines count as sum/initial. All-zero sequences leave count undetermined. A known constant endpoint alone also leaves count undetermined.
- Count inversion uses 64-digit decimal arithmetic. It can remain blank when a sum-inverse coefficient loses supplied digits at working precision, quadratic cancellation leaves insufficient root separation, a residual does not agree, or adjacent integer counts cannot be distinguished. A blank inverse is unresolved; supply a known count directly. This numerical limit does not cap supplied counts. Repeated quadratic roots are conservatively unresolved by this policy.
- Computed counts are checked against the original retained values using exact rational arithmetic. Verification supports up to65,536 text characters,32,768 coefficient/integer digits, exponent magnitude20,000,2,048 arithmetic steps and2,000,000 cumulative digit-work units. Exceeding these computation budgets leaves the inverse unresolved; enter a known count instead. These limits apply to inverse verification, not supplied counts or a physical sequence bound.
- At one term, final value and sum equal initial value mathematically; the difference is undetermined. If difference is missing, captured solver dependencies can leave these outputs blank. Some supplied combinations have no positive integer count. Directly supplied fractional counts are rejected. Computed count inverses remain subject to the disclosed64-digit resolution checks.
Common questions
Why is a list missing above 200terms?
The source limits the displayed list to200terms. The sum and final term are still calculated for larger valid counts.
Can a decreasing sequence have two counts with the same sum?
Yes. Starting at 5 with difference −1 gives sum 12 after 3 or 8 terms. The retained inverse returns 3; enter a known count directly to select the intended sequence.
References
Handle constant-sequence count inversion explicitly. Retain captured plus-square-root count when it is a positive integer; otherwise allow the sole admissible alternate root. Multiple positive counts remain branch-specific and disclosed. Zero-difference endpoint inversion is underdetermined, not a count of one. Reassociate all eight scalar variants to avoid the demonstrated large-term cancellation. Use conjugate quadratic roots while preserving the captured plus-root priority. Count inversion fails closed when discriminant cancellation is within1e-60 of its term scale or integer neighbors cannot be distinguished at64-digit working precision; this is a numerical resolution policy, not a magnitude bound on supplied counts. A finite number of sequence terms is a positive integer. Initial values, differences, endpoints and sums retain signed/zero domains. Counts above200 remain valid for analytic results.