Finite series partial sum

A finite series is a sum of a fixed number of terms written from a clear rule for the general term a_k. This capability accepts an explicit formula family, the inclusive index bounds, and the formula parameters, then adds each term in order and returns the partial sum together with first and last terms, the term count, and a short description of a_k.

What a partial sum of a finite series means

Writing S equals the sum from k equals from to k equals to of a_k makes the bounds explicit: every integer index in that closed interval contributes exactly one term, and the order of addition does not change the exact rational or real total for ordinary arithmetic. An explicit term formula tells you a_k without needing recursion: arithmetic sequences use a constant difference so a_k equals a plus (k minus from) times d; geometric sequences use a constant ratio so a_k equals a times r to the power (k minus from); polynomial formulas cover many textbook drills of the form c0 plus c1 times k plus c2 times k squared plus c3 times k cubed.

How this capability computes the answer

Arithmetic requires finite numbers a and d; geometric requires finite a and r; polynomial requires at least one of c0, c1, c2, or c3 as finite numbers, with omitted polynomial coefficients treated as zero. Geometric powers for moderate ranges are built by successive multiplication so the same inputs always produce the same floating-point path. You send a JSON body with a required formula string, arithmetic, geometric, or polynomial, plus integer from and to bounds with to greater than or equal to from. The engine accepts JSON numbers or plain numeric strings for coefficients, rejects missing required fields, booleans, objects, non-integers for indices, inverted ranges, and spans larger than ten thousand terms with invalid_input, and never charges those validation failures. It evaluates a_k at each index by the chosen explicit rule, adds successive terms into a running total, and returns formula, from, to, term_count, term_formula, first_term, last_term, partial_sum, the parameters that applied, and a terms array when the span has at most fifty terms. There is no randomness, no clock, and no network; identical inputs always yield identical JSON. The work is a short validation pass plus an O(n) accumulation loop, which sits inside the analytic cost model for this tier-A parse capability priced at $0.002 per successful request.

Where to use it and what it costs

The browser path runs locally and needs no account. Use the free browser tool for a single homework check or classroom demo, or call the API when a gradebook, tutor script, worksheet generator, or notebook must resolve many finite partial sums the same way every time. Typical placements include verifying student answers for S_n on arithmetic and geometric progressions, generating solution keys that list first and last terms beside the total, teaching how extending to changes the partial sum without rewriting the formula, and feeding a trusted partial_sum into a larger pipeline that already holds sequence parameters from a model or measurement. API usage draws from prepaid KIT balance at the published $0.002 base price per successful request; failed validation does not consume balance. Keep indices as integers, keep coefficients finite, prefer the server-computed partial_sum over re-adding with a different floating-point path in client code, and cap large ranges intentionally rather than summing millions of terms in a browser tab.