Sum of Series Calculator
Calculate arithmetic sums, finite geometric sums, and convergent infinite geometric sums from one consistent set of inputs.
Series inputs
Choose a constant-difference or constant-ratio model.
Required. Any finite decimal in U.S. notation, such as 1 or – 2.5.
Required. The fixed amount added from one term to the next.
Required for finite series. Enter a whole number from 1 to 1,000.
Live results
Sₙ = n/2 × [2a + (n – 1)d] = 100
Term-by-term breakdown
All 10 finite terms
| Index (k) | Term (aₖ) | Partial sum (Sₖ) | Share of final sum |
|---|
The table uses the same canonical values as the result cards and Excel workbook. Percentages are descriptive and may be negative or exceed 100% when terms change sign.
How to use the sum of series calculator
What this calculator does
This calculator finds three closely related quantities: the sum of a finite arithmetic series, the sum of a finite geometric series, or the limiting sum of an infinite geometric series that converges. It also identifies the last term of a finite series, the average term where that concept is finite, the convergence status, the exact formula used, and a term-by-term table. It is designed for structured arithmetic and geometric progressions; it does not evaluate arbitrary formulas, power series, or sequences whose difference or ratio changes from term to term.
When to use it
Use it to total evenly increasing payments or quantities, check a classroom sequence problem, model repeated growth or decay over a fixed number of steps, or estimate the limiting value of a repeating geometric process. For example, it can total the first ten odd numbers, project a balance multiplied by the same factor each period, or evaluate a convergent series such as 1 + 1/2 + 1/4 + 1/8 + ....
How to calculate
- The calculator opens with a complete demonstration: an arithmetic series with first term 1, common difference 2, and 10 terms. Its result and downloadable XLSX workbook are ready immediately.
- Choose Series type. Select Arithmetic series when consecutive terms differ by a constant, Finite geometric series when each term is multiplied by a constant ratio, or Infinite geometric series when you need the limiting sum.
- Replace the demonstration values in First term (a), Common difference (d) or Common ratio (r), and – when visible – Number of terms (n). Results update live.
- Read Series sum first, then verify Last term, Average term, Convergence, and Formula used. Review the term table to see how the partial sum develops.
- Select Download Excel to export the current validated inputs, outputs, and all generated rows. Reset clears the demonstration data rather than restoring it; after Reset, Excel export stays disabled until you enter another complete valid series.
Input guide
Series type is required and controls the mathematical model. Arithmetic uses a fixed difference; geometric uses a fixed ratio. Choosing the wrong type is the most common interpretation error. First term (a) is a required finite decimal with no unit imposed by the calculator; enter values such as 1, 250, or – 2.5 using a period as the decimal separator. A larger first term generally moves every term and the sum in the same direction, but sign-changing geometric ratios can produce oscillation. Scientific notation and decimal commas are intentionally rejected so values are never silently reinterpreted.
Common difference (d) appears for arithmetic series and is the fixed amount added each step. It is required, may be positive, zero, or negative, and a realistic example is 2 for the odd-number sequence. Increasing it raises later terms and usually raises the sum; a negative value creates a decreasing progression. Common ratio (r) appears for geometric series and is the fixed multiplier between terms. It is required and may be negative. For a finite example, 0.5 halves each term. For an infinite series, the strict domain is – 1 < r < 1; r = ±1 and |r| > 1 do not converge to a finite sum. OpenStax provides a detailed explanation of geometric sequences, finite sums, and convergent infinite series.
Number of terms (n) is required for finite series and hidden for an infinite geometric series. Enter a whole number from 1 through 1,000; for example, 10. It determines how many terms are included, so increasing it extends the sum by additional terms. Fractions, zero, negative counts, and formatted values such as “ten” are invalid. The 1,000-term cap keeps live rendering and workbook creation responsive while still covering typical educational and planning uses.
Output guide
Series sum is the primary result. For a finite series it is an exact algebraic identity for the supplied decimal inputs; for a convergent infinite geometric series it is the mathematical limit. Last term is the nth term for finite series and is shown as “Approaches 0” for a valid infinite geometric series. Average term equals the finite sum divided by n; it is not defined as a simple finite average for an infinite series, so the calculator reports “Not applicable” there. Convergence distinguishes a finite partial sum from a convergent infinite limit and warns when an infinite ratio is outside the required interval.
The header pills summarize Series, Terms, Last term, and Status using the same model values. Formula used shows the applicable identity with the current result. In the Term-by-term breakdown, Index (k) is the one-based position, Term (aₖ) is the current term, Partial sum (Sₖ) is the running total, and Share of final sum compares that running total with the final finite sum or infinite limit. A negative share or a value above 100% can be mathematically correct when alternating terms overshoot and then return toward the limit.
Worked example
The startup example is the arithmetic series 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19. Here a = 1, d = 2, and n = 10. The last term is aₙ = a + (n – 1)d = 1 + 9 × 2 = 19. The sum is Sₙ = n/2 × [2a + (n – 1)d] = 10/2 × [2 + 18] = 5 × 20 = 100. The average term is 100 ÷ 10 = 10. These values match the first-open cards, table, and workbook checkpoints. OpenStax derives the arithmetic-series sum formula from the first and last terms.
How the formulas work
For an arithmetic series, the nth term is aₙ = a + (n – 1)d and the sum is Sₙ = n(a + aₙ)/2. The equivalent expanded form used in the result panel is:
Sₙ = n/2 × [2a + (n – 1)d]
For a finite geometric series with r ≠ 1, the calculator uses Sₙ = a(1 – rⁿ)/(1 – r). When r = 1, every term equals a, so the stable special case is Sₙ = an. For an infinite geometric series, a finite limit exists only when |r| < 1, and then S = a/(1 – r). The Wolfram MathWorld geometric-series reference gives additional identities and notation.
Interpretation and common mistakes
Do not confuse a sequence with a series: a sequence lists terms, while a series adds them. Do not substitute a common difference into a geometric formula or a common ratio into an arithmetic formula. For infinite geometric calculations, “infinite” describes the number of terms, not an infinite answer; the limit is finite only when the absolute ratio is below one. With negative ratios, terms alternate signs and partial sums can cross the final limit repeatedly, so the table is often more informative than a single rounded result. Keep enough decimal precision in your inputs, and treat displayed rounding as presentation rather than a change to the underlying calculation.