Arithmetic Sequence Calculator - nth Term and Sum Finder

An arithmetic sequence calculator finds any value you need from a list of numbers that grows by a fixed step. Give it the first term a₁, the common difference d, and the number of terms n, and it returns the n-th term, the sum, the arithmetic mean, and a preview of the term list.

• • Find the n-th term of a progression: Type a₁, d, and n to read aₙ = a₁ + (n − 1) d without retyping the formula.
• • Sum a long arithmetic series: Add up the first n terms with Sₙ = n/2 (a₁ + aₙ) for bill totals or amortization.
• • Check homework or a textbook answer: Compare your work on 2, 5, 8, 11, ... at term 10 by reading the same aₙ and Sₙ the calculator prints.
• • Model a real-world linear plan: Use a positive d for a savings plan that adds the same deposit each month, or a negative d for a depreciation schedule.

An arithmetic sequence is a list of numbers where the difference between any two consecutive terms is the same constant d. That property is what makes the closed-form formulas work, and it is also what separates an arithmetic sequence from a geometric sequence, where each term is multiplied by a fixed ratio.

The calculator computes the n-th term, the sum, the mean, and the term list in one pass so you do not have to re-enter the same three numbers into separate tools.

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

Internally the calculator uses two closed-form formulas. The n-th term comes from aₙ = a₁ + (n − 1) d, and the sum of the first n terms comes from Sₙ = n/2 (a₁ + aₙ). The mean is just (a₁ + aₙ)/2, and the preview list loops the step d from a₁ to aₙ.

• a₁: First term. Any real number is allowed.
• d: Common difference. Positive d grows, negative d shrinks, d = 0 is constant.
• n: Number of terms. Must be a positive integer; preview list is capped at 50.
• aₙ: The n-th term, equal to a₁ + (n − 1) d. Used as the upper bound of the sum formula.
• Sₙ: Sum of the first n terms, equal to n/2 (a₁ + aₙ).

MathWorld publishes the n-th term and the sum formula as the two defining identities of an arithmetic progression, and Wikipedia repeats Sₙ = n/2 (a₁ + aₙ) because it falls out of pairing the first and n-th term. The calculator uses both formulas in closed form, so it never accumulates floating-point error for large n.

The mean row is the per-term average Sₙ/n.

According to Wolfram MathWorld, the n-th term of an arithmetic progression with first term a₁ and common difference d is aₙ = a₁ + (n − 1) d, and the sum of the first n terms equals n/2 (2 a₁ + (n − 1) d)

According to Wikipedia (Arithmetic progression), the sum of the first n terms of an arithmetic progression is Sₙ = n/2 (a₁ + aₙ), where aₙ is the n-th term of the sequence

When the values you have are real measurements rather than a closed-form step, Linear Regression Calculator fits a best-fit line through the same x and y pairs and returns a slope in the same units as aₙ.

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

The closed-form formulas are what separate an arithmetic sequence from a list of numbers. The moment d is constant, the two formulas above are exact, and the calculator can compute a₁₀₀ or S₁₀₀₀ in a single pass with no recursion.

The mean row is a sanity check: if the visible average of the preview list is not close to the printed mean, the typed-in d is wrong.

Because the common difference behaves like the slope of a linear function, Slope Percentage Calculator turns the same d into a percent rise so you can compare two progressions on a single scale.

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

1. 1 Type the first term: Enter a₁. Decimals, negatives, and fractions are all accepted.
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. The preview list is capped at 50 terms but the exact nth term and sum are still computed for larger n.
4. 4 Read the n-th term and the sum: The n-th term row is aₙ and the Sum row is Sₙ. 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 month 6 is 11,400 miles and the six-month total is 69,000.

The calculator removes the most common arithmetic-sequence mistakes and saves the step of looking up the formulas in a textbook.

• • All four answers in one pass: n-th term, sum, mean, and the term list print at the same time, so the arithmetic sequence calculator removes the need to recompute the same a₁ and d in a second tool.
• • Closed-form formulas, not loops: The calculator uses aₙ = a₁ + (n − 1) d and Sₙ = n/2 (a₁ + aₙ) directly, so the result is exact and does not accumulate floating-point error for large n.
• • Decreasing 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 is the same 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 the slope percentage or linear regression workflows.

If you are working through a problem set and bouncing between the n-th term and the sum, 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.

If the common difference is best described as a percent change rather than a flat step, Percentage Increase Calculator converts the same growth into a per-term percent that is easier to compare against another series.

Three things change the answer you should expect, plus two practical caveats about how arithmetic 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 arithmetic sequence model is only an approximation.
• • The calculator reports the closed-form n-th term, not the recursive definition. It cannot tell you the n-th term of a sequence where a non-constant rule kicks in at some index, such as arithmetic for the first 10 terms and geometric afterwards.

When you copy the sum into a spreadsheet or a budget tool, double-check the unit context. Sₙ 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 Khan Academy, an arithmetic sequence is a list of numbers in which the difference between consecutive terms is constant, and that constant is called the common difference d

When you want the other summary statistics for the same term list, Mean Median Mode Range Calculator computes the median, mode, and range in one pass so you can decide whether the mean of a₁ and aₙ is the right center to report.