Sum Of Linear Number Sequence Calculator - Closed-Form S_n Finder

A sum of linear number sequence calculator adds up the first n terms of an arithmetic progression, which is the textbook name for a linear number sequence. Give it the first term a₁, the common difference d, and the number of terms n, and it returns the closed-form sum S_n, the last term a_n, the mean of the first and last term, and a preview of the first terms so you can sanity check the step. It is the same identity behind the sum of an arithmetic series written n/2 (a₁ + a_n), so you do not have to add up the terms by hand.

• • Sum a long arithmetic series: Read S_n in one pass without retyping the terms or pulling out a calculator for every pair.
• • Total a savings plan with a fixed deposit: Model a weekly or monthly deposit that grows by a constant amount each period.
• • Check a homework or textbook answer: Compare your work on 2, 5, 8, 11, ... at term 10 against the same closed-form sum the calculator prints.
• • Total a depreciation schedule: Use a negative d for a steady monthly loss, the same way you would total an increasing series.

Because both formulas are closed form, the calculator returns the sum exactly for any n in the allowed range and does not accumulate rounding error the way a recursive loop would.

If you also need the nth term or a full term list for the same progression, the Arithmetic Sequence Calculator sits one click away and shares the same inputs.

Internally the calculator uses the closed-form identity S_n = n/2 (a₁ + a_n). When you only know a₁ and d, it first computes a_n = a₁ + (n - 1) d and then plugs that value into the sum formula. The mean and the average-term rows are the same number in two forms because the step is constant.

• a₁: First term of the linear number sequence. Any real number is allowed.
• d: Common difference between consecutive terms. Positive d grows the series, negative d shrinks it, d = 0 is constant.
• n: Number of terms to sum. Must be a positive integer between 1 and 1000.
• a_n: Last term, equal to a₁ + (n - 1) d. Used as the upper bound of the sum formula.
• S_n: Sum of the first n terms, equal to n/2 (a₁ + a_n).

The mean of the endpoints equals S_n / n whenever the step is constant, which is why the calculator prints both values: if they disagree, the typed-in d is wrong.

According to Wolfram MathWorld, the sum of the first n terms of an arithmetic progression with first term a_1 and common difference d is S_n = n/2 (2 a_1 + (n - 1) d)

When the inputs you have are real measurements rather than a closed-form step, Linear Regression Calculator fits the same linear pattern through your data and returns a slope in the same units.

Four short ideas are enough to use the closed-form sum correctly on any linear number sequence.

The closed-form formulas are what separate a linear number sequence from an arbitrary list. The moment d is constant, both forms are exact, and the calculator can compute S_1000 in a single pass with no recursion.

The mean of the endpoints and the average term S_n / n are the same number, and that is the sanity check: if the two values disagree in the result panel, the typed-in d is wrong.

For a sequence that does not have a constant step, Sequence Calculator handles the more general case and still returns the first n terms.

Five short steps cover every workflow this calculator supports, from a single homework problem to a long savings plan.

1. 1 Type the first term: Enter a₁. Decimals, negatives, and fractions are all accepted because a linear number sequence can start anywhere.
2. 2 Type the common difference: Enter d, the step between consecutive terms. Use a positive number to grow, a negative number to shrink, and 0 for a constant sequence.
3. 3 Type the number of terms: Enter n as a positive integer between 1 and 1000. The preview list is capped at 50 terms, but the exact sum of linear number sequence is still computed for larger n.
4. 4 Read the sum S_n: The Sum row is the closed-form total of the first n terms. The Last term row is a_n, and the Mean row is the per-term average that gets multiplied by n to give the sum.
5. 5 Scan the term list: The first n terms row prints a comma-separated preview. Use it to confirm that d is the step you intended, and that the list is increasing, decreasing, or constant.

A driver logs 12,400 miles in January and then drives 200 fewer miles each subsequent month. Set a₁ = 12400, d = -200, and n = 6 to see that the six-month total is 69,000 miles and the month-6 value is 11,400 miles.

When the series you want to total is geometric or a mix of geometric and constant pieces, Sum of Series Calculator handles the more general case in the same math hub.

The sum of linear number sequence calculator removes the most common mistakes when adding a long arithmetic series by hand.

• • Closed-form sum, not a loop: S_n is computed directly with n/2 (a₁ + a_n), so the result is exact and does not accumulate floating-point error for large n.
• • All four answers in one pass: Sum, last term, mean, and the term list print at the same time, so you do not have to recompute the same a₁ and d in a second tool.
• • Decreasing and constant sequences handled the same way: Negative d is just another number, so depreciation schedules, pay-down plans, and any other decreasing series use the same workflow as increasing ones.
• • Domain errors are explained: If n is not a positive integer, the calculator shows a clear error message instead of returning a NaN, which is what most spreadsheet formulas do when they hit a bad input.
• • Pairs with related math tools: The mean row matches the value an average calculator prints for two inputs, and the common difference behaves like the slope of a linear function, so the result feeds into slope or regression workflows.

If you are working through a problem set and bouncing between the sum and the term list, the calculator removes the chance of mixing up the formulas.

The calculator is just JavaScript in the browser, so it updates as you type and tolerates negative steps, non-integer steps, and long lists.

For the same average computation on a list of numbers that are not equally spaced, Average Calculator handles the general case without requiring a constant step.

Three things change the answer you should expect, plus two practical caveats about how linear number sequences behave in the real world.

• • The formulas assume a constant step d. If your real-world series has a drifting step (such as a salary that grows by 3% one year and 4% the next), the linear number sequence model is only an approximation.
• • The calculator reports the closed-form sum, not the recursive definition. It cannot total a sequence where a non-constant rule kicks in at some index, such as linear for the first 10 terms and geometric afterwards.

When you copy S_n into a spreadsheet or a budget tool, double-check the unit context. S_n is a pure count, not a currency, so if the original terms are dollars you still have to multiply the sum by the unit you started with.

If the sum looks correct but does not match the textbook, the textbook may use 1-based indexing with a different a₁. Walk the preview list against the problem statement to make sure n is the count the textbook uses.

According to Wikipedia (Arithmetic progression), the sum of the first n terms of an arithmetic progression is S_n = n/2 (a_1 + a_n), which follows from pairing the first and n-th term

According to Omnicalculator, the sum of a linear number sequence is the closed-form sum of an arithmetic progression given the first term, the common difference, and the number of terms

When you want the median, mode, and range for the same term list, Mean Median Mode Range Calculator computes those summary statistics in one pass.