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Summation of Series Calculator | Sum arithmetic series from a,d,n

Sum arithmetic and geometric series with the nth-term formula and the sum formula shown step by step. Enter the first term, difference or ratio, and the number of terms — ideal for sequences homework and financial projections.

Last updated: June 2026 · Free · No sign-up required

Results

Enter values above and click Calculate to see your result instantly.

Inside the Summation Of Series Calculator

Under the hood, this calculator applies aₙ = a + (n−1)d; Sₙ = n(a + aₙ)÷2 = n÷2 × (2a + (n−1)d), so the summation of series calculator result you see is genuine math, not a lookup table.

The reference table below covers the most common summation of series calculator cases at a glance, the methodology section breaks the calculation into verifiable steps, and the FAQ tackles the edge cases.

Quick reference

SeriesSum
1+2+…+100 (a=1, d=1, n=100)5,050
2+5+8+… (a=2, d=3, n=10)155
5+10+…+50 (a=5, d=5, n=10)275
Odd numbers 1+3+… (n=10)100

info Summation Of Series Calculator

Free math calculator — enter your numbers and get an instant, accurate result.

info Private by design

Everything runs locally in your browser. No uploads, no accounts, no tracking.

info Works everywhere

Fully responsive and mobile-friendly — calculate on any device, any time.

info Educational

Includes the formula and step-by-step explanation so you understand the math, not just the answer.

Interpretation guide

QuantityMeaning
aₙ (last term)a + (n−1)d — where the series ends
Average term(a + aₙ)÷2 — the series’ midpoint
Sumn × average term

lightbulb Worked example

Let's say you are using the Summation Of Series Calculator. Calculates summation of series using aₙ = a + (n−1)d, in real time in your browser. Enter the values that match your situation into the input fields and press calculate — using realistic numbers makes the result directly useful for you.

Result: The calculator instantly applies the formula aₙ = a + (n−1)d; Sₙ = n(a + aₙ)÷2 = n÷2 × (2a + (n−1)d) and returns the result with appropriate precision.

What this means: Read the result in the context of what you are measuring. The step-by-step breakdown lets you confirm the math and understand which input most affects the outcome.

Formula & methodology

Formula: aₙ = a + (n−1)d; Sₙ = n(a + aₙ)÷2 = n÷2 × (2a + (n−1)d)

The Summation Of Series Calculator is built on a well-established calculation method. It uses the formula aₙ = a + (n−1)d; Sₙ = n(a + aₙ)÷2 = n÷2 × (2a + (n−1)d) to turn your inputs into a reliable result. Calculates summation of series using aₙ = a + (n−1)d, in real time in your browser. The steps are shown on the page so you can follow the reasoning from input to output.

  1. Enter the first term a, the common difference d and the number of terms n.
  2. The last term is aₙ = a + (n−1)d.
  3. The sum uses Gauss’s pairing trick: S = n × (a + aₙ) ÷ 2.

Example: a = 2, d = 3, n = 10 → last term 29, sum = 10 × (2+29)÷2 = 155.

Authoritative source: Wolfram MathWorld

Frequently asked questions

What is the formula for the sum of an arithmetic series?
The Summation Of Series Calculator uses the standard aₙ = a + (n−1)d; Sₙ = n(a + aₙ)÷2 = n÷2 × (2a + (n−1)d). You enter your values, and the calculator applies the formula step by step, showing the working so you can verify the math and understand how the result is derived rather than trusting a black box.
What is 1 + 2 + … + 100?
The Summation Of Series Calculator is free, private, and accurate: it runs entirely in your browser (no uploads, no accounts), applies the standard calculation, and explains each step so you can verify the result. Calculates summation of series using aₙ = a + (n−1)d, in real time in your browser. There is no limit on usage, and it works on any device.
What is a common difference?
The common difference d is the constant amount added between consecutive terms of an arithmetic series — in 2, 5, 8, 11, the common difference is 3. It defines the sequence along with the first term: term n = a₁ + (n−1)d. The calculator uses it to generate terms and sum the series with Sₙ = n(a₁ + aₙ) ÷ 2.
Does this work for geometric series?
Yes — the calculator also handles geometric series, where each term multiplies by a constant ratio r (like 2, 6, 18, 54 with r = 3). The sum uses Sₙ = a₁(1 − rⁿ) ÷ (1 − r) for finite series, and infinite geometric series converge to a₁ ÷ (1 − r) when |r| < 1. Select the series type and the tool applies the matching formulas.
Can the difference be zero?
Yes — a common difference of zero makes every term identical: 7, 7, 7, 7 is a valid arithmetic sequence with d = 0. The sum is simply n × the term value, which the formulas handle correctly (the division by (1−r) in the geometric sum would be undefined for r = 1, but the calculator detects that edge case and returns n × a instead).
What does the Summation Of Series Calculator do?
Calculates summation of series using aₙ = a + (n−1)d, in real time in your browser. The calculator takes your inputs, applies the standard calculation, and returns a clear result so you can make an informed decision without doing the math by hand.
What formula does the Summation Of Series Calculator use?
The Summation Of Series Calculator applies the standard formula: aₙ = a + (n−1)d; Sₙ = n(a + aₙ)÷2 = n÷2 × (2a + (n−1)d). The tool walks through each step of the calculation so you can verify the numbers yourself.
Is the Summation Of Series Calculator free and private?
Yes — the Summation Of Series Calculator. There is no sign-up, no paywall, no trial, and no limit on how many calculations you can run. CalcaTools covers its costs with non-intrusive display advertising, so the calculator itself never asks for payment or restricts any feature.
How accurate is the Summation Of Series Calculator?
The Summation Of Series Calculator applies the standard formula: aₙ = a + (n−1)d; Sₙ = n(a + aₙ)÷2 = n÷2 × (2a + (n−1)d). It is tested against published worked examples before launch and uses native double-precision arithmetic, so results are accurate to typical precision for the values you enter — with no intermediate rounding that could distort the answer.

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