Arithmetic Series Calculator

Free Online Arithmetic Series Calculator

This arithmetic series calculator finds the sum of an arithmetic series (Sₙ) instantly, and it can also work backward to find the first term, the common difference, or the number of terms if you already know the total sum. Enter the values you know, and the calculator rearranges the arithmetic series formula for you, shows a labeled waterfall accumulation diagram, and lays out the complete step-by-step solution behind the answer — so you never have to guess which formula to use or how to rearrange it by hand.

An arithmetic series is simply the sum of the terms of an arithmetic sequence. If the sequence is 3, 8, 13, 18, 23, 28, 33, 38, 43, 48, then the arithmetic series is the running total you get by adding those terms together: 3 + 8 + 13 + 18 + 23 + 28 + 33 + 38 + 43 + 48 = 255. This calculator handles that addition instantly for any number of terms, no matter how large, using the closed-form sum formula instead of adding term by term.

Arithmetic Series Formula

Every calculation on this page traces back to one core arithmetic series formula, which can be rearranged to solve for whichever variable is missing:

• Sum formula: Sₙ = n/2 × (2a₁ + (n − 1)d)
• Equivalent sum formula (when the last term is known): Sₙ = n/2 × (a₁ + aₙ)
• Solving for the first term: a₁ = Sₙ/n − (n − 1)d/2
• Solving for the common difference: d = (2Sₙ/n − 2a₁) / (n − 1)
• Solving for the number of terms: n is the positive whole-number root of d·n² + (2a₁ − d)·n − 2Sₙ = 0

How to Use This Arithmetic Series Calculator

Pick what you want to solve for from the dropdown at the top: the sum, the first term, the common difference, or the number of terms. The input fields automatically relabel to show only the values you need for that mode, and every result updates the moment you type — the first term, common difference, number of terms, last term, and the sum of the series all appear together in the results panel.

Below the results, the waterfall accumulation diagram renders the series as a sequence of stacked columns. Each column's grey base is the total carried forward from every earlier term, and the colored slice stacked on top is the exact value of the new term being added at that step — so the running total, labeled directly on the diagram as Σ=, climbs visibly with every column until the last bar shows the full series sum. Underneath that, the step-by-step solution shows exactly how the answer was derived, formula by formula, from your entered numbers.

Worked Example: Finding the Sum of a Series

Suppose a₁ = 3, the common difference d = 5, and you want the sum of the first 10 terms. First find the last term: a₁₀ = a₁ + (n − 1)d = 3 + (10 − 1) × 5 = 3 + 45 = 48.

Then apply the sum formula: Sₙ = n/2 × (a₁ + aₙ) = 10/2 × (3 + 48) = 5 × 51 = 255. So the series 3 + 8 + 13 + 18 + 23 + 28 + 33 + 38 + 43 + 48 adds up to exactly 255 — the same numbers this calculator's default example uses, and the same total you can see building up column by column in the diagram above.

Worked Example: Finding the First Term from the Sum

Now suppose you're told a series has a common difference of d = 4, runs for n = 12 terms, and adds up to a total of Sₙ = 300, but you don't know where it started. Rearranging the sum formula gives a₁ = Sₙ/n − (n − 1)d/2 = 300/12 − (12 − 1) × 4/2 = 25 − 22 = 3.

So the series begins at a₁ = 3 and increases by 4 each time: 3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 43, 47 — and adding all twelve of those terms does indeed total 300. Switching this calculator's 'Solve for' dropdown to First Term and entering d = 4, n = 12, and Sₙ = 300 reproduces this exact result along with the full working.

Worked Example: Finding the Common Difference from the Sum

Suppose a series starts at a₁ = 5, runs for n = 8 terms, and the sum of those 8 terms is Sₙ = 148, but the common difference is unknown. Rearranging gives d = (2Sₙ/n − 2a₁) / (n − 1) = (2 × 148/8 − 2 × 5) / (8 − 1) = (37 − 10) / 7 = 27/7 ≈ 3.857.

This confirms that the common difference doesn't have to be a whole number — fractional and decimal common differences are perfectly valid, and this calculator handles them with the same precision as whole-number cases.

Worked Example: Finding the Number of Terms from the Sum

Finally, suppose a₁ = 2, the common difference d = 3, and the series sums to Sₙ = 345, but you don't know how many terms were added. Substituting into d·n² + (2a₁ − d)·n − 2Sₙ = 0 gives 3n² + (4 − 3)n − 690 = 0, or 3n² + n − 690 = 0.

Solving this quadratic with the quadratic formula produces n = 15 as the only positive whole-number root (the other root is negative and gets discarded automatically). Checking the answer: Sₙ = 15/2 × (2 × 2 + (15 − 1) × 3) = 7.5 × (4 + 42) = 7.5 × 46 = 345, which matches exactly. This is why the calculator always checks the discriminant and rounds carefully, flagging an error whenever a target sum doesn't land on a clean whole-number number of terms instead of returning a misleading fractional answer.

Understanding the Waterfall Accumulation Diagram

The diagram is built so the sum makes visual sense without reading a single row of numbers as text. Every column has two parts stacked on top of each other: the grey base represents everything already added before that step, and the colored cap on top — amber for the first term, indigo for the middle terms, green for the final term — is the exact size of the new term being added right then, labeled directly with its value as '+value'.

Beneath each column, a bold 'Σ=' label shows the running total after that term is added, so you can watch the sum grow one column at a time and see exactly which addition pushed the total from one number to the next. The final column's total — highlighted in green — is always the answer shown as Sₙ in the results panel. For series with many terms, the diagram compresses the middle into an ellipsis so the first few terms and the final term stay clearly visible without overcrowding the chart.

Arithmetic Series vs. Arithmetic Sequence

It's easy to mix these two terms up, but the distinction is simple: an arithmetic sequence is the ordered list of individual terms (2, 5, 8, 11), while an arithmetic series is what you get when you add those terms together (2 + 5 + 8 + 11 = 26). If you only need to find an individual term — like the 20th number in the pattern — rather than a total, use the dedicated Arithmetic Sequence Calculator instead; this page is specifically built around the sum.

Both tools share the same underlying common difference and first-term logic, so values you calculate on one page will always be consistent with the other — a series calculator is really just a sequence calculator with one extra step, the addition of every term.

Real-World Applications of Arithmetic Series

Arithmetic series show up anywhere a fixed, predictable pattern needs to be totaled rather than just listed:

• Total savings after a fixed number of deposits that each increase by the same amount, such as saving ₹500 more every month than the month before.
• Total seating capacity of a stadium or theater where each row has a constant number of extra seats compared to the row before it.
• Total number of objects stacked in a pyramid arrangement, like crates, logs, or cans, where each layer changes by a fixed count.
• Total distance traveled under constant acceleration, summing the distance covered in each successive time interval in a physics problem.
• Total interest paid across a fixed number of periods under simple interest, where each period's payment increases (or decreases) by the same amount.
• Total cost or total production across a set number of periods when a quantity grows or shrinks by a constant amount each period, such as a factory increasing daily output by a fixed number of units.

Tips for Working with Arithmetic Series

Always find (or confirm) the last term aₙ before applying Sₙ = n/2 × (a₁ + aₙ) — it's often the fastest route to the sum once you already know the first and last terms of a finite series. If you only know a₁, d, and n, the expanded formula Sₙ = n/2 × (2a₁ + (n − 1)d) skips that extra step and gets you straight to the answer.

When solving for the number of terms from a target sum, remember the relationship becomes a quadratic equation, so there can be two mathematical roots — but only a positive whole number makes sense as an actual count of terms, and this calculator automatically discards the invalid root for you. If neither root is a clean positive integer, that's a strong signal the target sum you entered doesn't actually correspond to a valid whole-number-length series with the given a₁ and d.