Sum Of Series Calculator - Arithmetic and Geometric Sums

A sum of series calculator finds the partial sum of the first n terms of a series without adding them up by hand. Switch the series type to arithmetic or geometric, give it the first term a₁, the step (d or r), and n, and it returns the n-th term, the partial sum S_n, the mean, and a term list preview.

• • Sum an arithmetic or geometric series: Type a₁, the step, and n to read S_n = n/2 (a₁ + a_n) for arithmetic or S_n = a₁ (1 - r^n) / (1 - r) for geometric.
• • Find the n-th term of a series: Use a_n = a₁ + (n − 1) d for arithmetic or a_n = a₁ r^(n − 1) for geometric to read a late term directly.
• • Check a homework or textbook answer: Compare your work on 2, 5, 8, 11, ... at n = 10 or 1, 2, 4, 8, ... at n = 8 by reading the same S_n the calculator prints.
• • Model a real-world growth or decay plan: Use a positive step for an arithmetic deposit plan, a geometric growth model, or a ratio between 0 and 1 for a decay schedule.

An arithmetic series adds d, and a geometric series multiplies by r. Both have a closed-form partial sum that the calculator reads in a single computation.

When you only need the arithmetic branch with the same a₁ and d inputs, the Arithmetic Sequence Calculator focuses the workflow on the arithmetic closed-form formulas and returns the same a_n and S_n in the same units.

Internally the calculator uses two closed-form formula families. For arithmetic series the n-th term is a_n = a₁ + (n − 1) d and the partial sum is S_n = n/2 (a₁ + a_n). For geometric series the n-th term is a_n = a₁ r^(n − 1) and the partial sum is S_n = a₁ (1 − r^n) / (1 − r) when r is not 1, with the special case S_n = n a₁ when r = 1.

• seriesType: Arithmetic or geometric. Sets which second input is used and which formula runs.
• a₁: First term. Any real number is allowed.
• d: Common difference for arithmetic series. Positive d grows, negative d shrinks, d = 0 is constant.
• r: Common ratio for geometric series. The closed-form sum is not valid at r = 1; the calculator falls back to S_n = n a₁.
• n: Number of terms. Must be a positive integer; the preview list is capped at 50 terms.

MathWorld publishes a_n and S_n as the two defining identities of an arithmetic progression, and Wikipedia gives the matching closed-form partial sum for a geometric series. The calculator uses both formula families in closed form, so it skips the term-by-term loop that would compound round-off; the final result is still IEEE 754 double precision, so very large n or extreme r values can round the last few digits.

According to Wolfram MathWorld, an arithmetic progression gives a_n = a₁ + (n − 1) d and S_n = n/2 (2 a₁ + (n − 1) d)

According to Wikipedia (Geometric series), a geometric series has partial sum S_n = a (1 − r^n) / (1 − r) when r ≠ 1, and S_n = n a when r = 1

When you only need the geometric branch with the same a₁ and r inputs, the Geometric Sequence Calculator focuses the workflow on the geometric closed-form formulas and walks the same r^n and S_n term by term.

These four ideas are enough to use the calculator correctly on any arithmetic or geometric input.

The closed-form formulas are what separate a series from a list of numbers. The moment the step is constant, the two formulas above are exact.

For a sum of trigonometric expressions where the closed-form partial sum is built from sin and cos identities, the Sum Difference Identities Calculator handles the trig branch with the same n and term pattern as this calculator.

Five short steps cover every workflow the sum of series calculator supports, from a homework problem to a long growth plan.

1. 1 Pick the series type: Choose arithmetic if the series adds a constant step, or geometric if it multiplies by a constant ratio.
2. 2 Type the first term: Enter a₁. Decimals, negatives, and fractions are all accepted for both series types.
3. 3 Type the step (d or r): For arithmetic, enter d. For geometric, enter r. Use a positive number to grow, a negative number for alternating sign, and 0 or 1 for a constant series.
4. 4 Type the number of terms: Enter n as a positive integer. The preview list is capped at 50 terms but the exact a_n and S_n are still computed for larger n.
5. 5 Read a_n and the partial sum: The n-th term row is a_n and the partial sum row is S_n. The mean row is the per-term average S_n / n, and the term list prints a comma-separated preview.

A small business forecasts revenue: arithmetic growth of $1,200 per quarter for 8 quarters (a₁ = 48000, d = 1200, n = 8), then geometric growth at 1.05 per quarter for the next 8 (a₁ = 57600, r = 1.05, n = 8).

The calculator removes the most common sum-of-series mistakes and saves the step of looking up two different formulas in a textbook.

• • Both series types in one tool: Arithmetic and geometric partial sums use the same form, so one calculator handles a homework set, a depreciation schedule, and a growth model without switching tools.
• • Closed-form formulas, not loops: The calculator uses a_n = a₁ + (n − 1) d, S_n = n/2 (a₁ + a_n), a_n = a₁ r^(n − 1), and S_n = a₁ (1 − r^n) / (1 − r) directly, so a_n and S_n are computed in a fixed number of operations rather than n additions; the result is still IEEE 754 double precision, so very large n or extreme r values can round the last few digits.
• • Handles the r = 1 corner case: The closed-form sum formula divides by (1 − r) and is undefined at r = 1; the calculator falls back to S_n = n a₁ automatically.
• • Domain errors are explained: If n is not a positive integer, or a term overflows JavaScript number precision, the calculator shows a clear error message instead of returning NaN or Infinity.
• • Pairs with related math tools: The mean row matches the average calculator's output, and the n-th term is the seed for any sequence-style extension.

When the series you need to sum is the digits of an integer rather than a constant-step arithmetic or geometric progression, the Sum Of Digits Calculator adds the digits of any number and reports the same kind of partial sum the closed-form formula would give.

Three things change the answer you should expect, plus two practical caveats about how real-world series behave outside the closed-form model.

• • The formulas assume a constant step. If the real-world series has a drifting step (such as a salary that grows by 3% one year and 4% the next), the sum-of-series model is only an approximation.
• • The calculator reports the partial sum of the first n terms, not the limit as n goes to infinity. For geometric series with |r| < 1 the infinite sum is a₁ / (1 − r); this calculator is built for finite partial sums.

When you copy the partial sum into a spreadsheet, the unit is whatever the terms already carry. Dollar terms in give dollar sums out, meter terms in give meter sums out, dimensionless counts stay dimensionless; the calculator does not convert or rescale the unit, so a quarterly revenue forecast in dollars and a depreciation schedule in years stay in their own units when summed.

According to Khan Academy, a geometric sequence multiplies each term by a fixed, non-zero common ratio r

When you want the mean of the same term list without the sum-of-series framing, the Average Calculator computes the arithmetic mean of a set of numbers in a single pass and reports the same value as the S_n / n row in this calculator.