Sequence Calculator - nth Term, Sum, and Term List Finder

A sequence calculator is a single tool that produces the nth term, the sum of the first n terms, the mean of those n terms, and a preview of the term list for any of the three main kinds of number sequences used in algebra. Pick the sequence type, enter the first term and the step or ratio, and the page returns the closed-form answers in one pass.

• • Find the n-th term of a sequence: Type the first term, the step or ratio, and the term index, then read a_n without walking through every intermediate term by hand.
• • Sum a long arithmetic or geometric series: Get S_n in closed form for a long sum such as the first 50 terms of a doubling geometric series or the first 100 terms of a savings plan.
• • Check a textbook or homework answer: Compare your work on 2, 5, 8, 11, ... or 3, 6, 12, 24, ... against the same a_n and S_n the calculator prints at the bottom of the form.
• • Model real-world linear or compounding plans: Use a positive d for a savings plan that adds the same deposit each month, or r > 1 for a growth series that compounds at a fixed rate per period.

For the arithmetic-only workflow that prints the same outputs from a_1, d, and n, the Arithmetic Sequence Calculator is a faster fit.

Internally the calculator switches between three closed-form formulas based on the sequence type, computes the n-th term, the sum, the mean, and walks the first 50 terms to build a preview list.

• type: Sequence type. arithmetic uses d, geometric uses r, fibonacci ignores a_1 and d and starts at F_1 = F_2 = 1.
• a_1: First term. Any real number for arithmetic and geometric; hidden for Fibonacci.
• d: Common difference for arithmetic sequences. Positive d grows, negative d shrinks, d = 0 is constant.
• r: Common ratio for geometric sequences. r = 1 is constant, r = 0 collapses to zero after a_1, |r| < 1 shrinks toward zero.
• n: Term index. Must be a positive integer; preview list is capped at 50 terms.

For arithmetic and geometric sequences the calculator uses the closed-form n-th term and sum, so it never accumulates floating-point error.

According to Wikipedia (Geometric progression), the n-th term of a geometric progression is a_n = a_1 * r^(n - 1) and the sum of the first n terms, when r is not 1, is a_1 (1 - r^n) / (1 - r).

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

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

These four ideas are enough to pick the right formula, read the result, and explain the answer in your own words.

The choice between arithmetic and geometric is the most common decision. Ask whether the change from one term to the next is a fixed amount (arithmetic), a fixed percent (geometric), or the sum of the two previous terms (Fibonacci).

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

Five short steps cover every workflow the page supports, from a homework problem to a long growth projection.

1. 1 Pick the sequence type: Open the Sequence type dropdown and choose arithmetic, geometric, or fibonacci. The page shows the same form for all three and only uses the inputs that match the type.
2. 2 Enter the first term a_1: Type a_1 for arithmetic or geometric sequences. The Fibonacci option ignores this input and starts at F_1 = F_2 = 1.
3. 3 Enter the step or ratio: Type d for arithmetic sequences, r for geometric sequences. Use a positive value to grow, a negative value to shrink, and 0 or 1 to make a constant sequence.
4. 4 Enter the number of terms n: Type n as a positive integer. The preview list is capped at 50 entries, but the closed-form n-th term and sum are computed for any n in the allowed range.
5. 5 Read the n-th term, the sum, and the mean: The primary row is a_n. The Sum row is S_n in closed form. The Mean row is S_n divided by n, and the preview list shows the first 50 terms as a comma-separated string.

A savings plan adds 100 dollars in month 1 and 25 dollars more each subsequent month. Pick arithmetic, set a_1 = 100 and d = 25, and set n = 24 to see that month 24 is 675 dollars and the two-year total is 9,300 dollars.

For the median, mode, and range of the same term list, the Mean Median Mode Range Calculator returns them in one pass so you can decide whether the printed mean is the right center to report.

The page removes the most common sequence mistakes and saves the step of looking up three different formulas in three different references.

• • Three sequence types in one tool: Arithmetic, geometric, and Fibonacci answers come from the same form, so you do not have to retype the inputs into a separate Fibonacci calculator to compare rules.
• • Closed-form answers, not loops: The page uses a_n = a_1 + (n - 1) d, a_n = a_1 * r^(n - 1), and F_(n+2) - 1 directly, so the result is exact and does not accumulate floating-point error for large n.
• • Decreasing and alternating sequences handled the same way: Negative d and negative r are just another number, so depreciation schedules, pay-down plans, and sign-alternating geometric series use the same workflow as increasing ones.
• • Domain errors are explained: If n is not a positive integer, the page surfaces a clear error message instead of returning NaN, which is what most spreadsheet formulas do when they hit a bad input.
• • Visible preview list, not just the n-th term: The first 50 terms are printed as a comma-separated string so you can confirm that d or r is the step you intended before trusting the closed-form sum.

The page updates the result on every keystroke, so changing d, r, or n moves a_n and S_n at the same time, which is the fastest way to build intuition for the closed-form formulas.

If the common ratio is best described as a percent change rather than a flat multiplier, the 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 sequences behave in the real world.

• • The formulas assume the rule is constant across the whole list. If your real-world series has a drifting step or a drifting ratio, the sequence model is only an approximation of the actual numbers.
• • The calculator reports the closed-form n-th term, not the recursive definition, so it cannot tell you the n-th term of a sequence where the rule changes 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. S_n is a pure count, so if the original terms are dollars or another unit, you still have to multiply the sum by the unit you started with to recover a currency answer.

According to Wikipedia (Fibonacci sequence), the Fibonacci sequence is defined by F_1 = F_2 = 1 and F_n = F_(n-1) + F_(n-2) for n >= 3, and the sum of the first n Fibonacci numbers is the closed-form identity F_(n+2) - 1.

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