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Arithmetic, geometric & Fibonacci

Number Sequence Calculator

This calculator finds the n-th term and the sum of the first n terms for arithmetic, geometric, and Fibonacci sequences. Enter a first term and common difference or ratio (Fibonacci uses F₁ = F₂ = 1), pick target n up to 500, and read the Result panel plus step-by-step Logic Trace. It also shows a term preview bar, optional S∞ when |r| < 1, and copyable first 20 terms. It does not infer a pattern from a list you provide.

By Jeff Beem

Updated

Sequence Laboratory

Choose a sequence type and enter the parameters. The Analytical Dashboard on the right shows the n-th term, sum, and a visual trace.

1 to 500. For Fibonacci, large n may produce very large terms.

Result
an (n-th term)29
Sn (sum to n)155

Formula

aₙ = a₁ + (n−1)d = 2 + (10−1)×3 = 29

First 10 terms (preview)

Preview shows up to 10 terms; copy exports the first 20.

Logic Trace

StepDetail
Formulaaₙ = a₁ + (n−1)d
Substituteaₙ = 2 + (10−1)×3
Simplifyaₙ = 2 + 27 = 29
Sum SₙSₙ = n/2 × (2a₁ + (n−1)d) = 10/2 × (4 + 27) = 155

Reading your sequence results

A number sequence is an ordered list that follows a rule. Choose the rule that matches your problem, enter the starting values, and the panels on the right show both the term you asked for and the sum of all terms up to that point.

Example: arithmetic a₁=2, d=3, n=10 (defaults)

With Arithmetic selected and the default inputs (a₁ = 2, d = 3, n = 10), each term rises by 3: 2, 5, 8, 11, and so on. The tenth term is a₁₀ = 2 + (10−1)×3 = 29. Adding all ten terms gives S₁₀ = 10/2 × (2×2 + 9×3) = 155. The First 10 terms (preview) bar displays 2, 5, 8, …, 29 with +d between chips so you can see the steady step.

Formula and Last additions boxes

For arithmetic and geometric sequences, the Formula box rewrites the general rule with your numbers already plugged in, which is useful when you are checking homework. Fibonacci mode switches the heading to Last additions and shows the final few sums that build up to Fₙ (at n = 10, you will see lines like 13 + 21 = 34 leading to F₁₀ = 55).

Switch modes to compare

The same target index n can mean very different things depending on the pattern. Try Geometric with the defaults (a₁ = 2, r = 2, n = 10): the tenth term jumps to 1,024 and S₁₀ = 2,046. Switch to Fibonacci at n = 10 and you get F₁₀ = 55 with sum 143, which equals F₁₂ − 1. Toggling modes is an easy way to build intuition for how each rule behaves.

Number sequence calculator: n-th term and sum

Find any term and the sum of the first n entries in arithmetic, geometric, or Fibonacci sequences, with step-by-step work shown alongside the answer.

What this calculator returns

A number sequence is an ordered list of numbers that follow a pattern. Each entry is called a term, and the position of a term is its index n (the first term is n = 1, the second is n = 2, and so on). This page focuses on three patterns you meet early in algebra and precalculus: arithmetic sequences, where you add the same amount each time; geometric sequences, where you multiply by the same ratio each time; and the Fibonacci sequence, where each term after the first two is the sum of the two before it. You choose the pattern, enter the starting values, and the calculator returns the n-th term and the sum of the first n terms. It does not guess a pattern from a list you paste in; you tell it which rule to use.
  • Arithmetic (constant difference d):
    Each term is the previous term plus d. The n-th term is
    an=a1+(n−1)da_n = a_1 + (n-1)d
    The partial sum of the first n terms is
    Sn=n2(2a1+(n−1)d)S_n = \frac{n}{2}(2a_1 + (n-1)d)
  • Geometric (constant ratio r):
    Each term is the previous term times r. The n-th term is
    an=a1⋅rn−1a_n = a_1 \cdot r^{n-1}
    For a finite sum when r ≠ 1, use
    Sn=a1(1−rn)1−rS_n = \frac{a_1(1-r^n)}{1-r}
    When |r| < 1, the infinite sum converges to
    S∞=a11−rS_\infty = \frac{a_1}{1-r}
  • Fibonacci (on this page F₁ = F₂ = 1):
    For n ≥ 3, each term follows
    Fn=Fn−1+Fn−2F_n = F_{n-1} + F_{n-2}
    The sum of the first n terms equals
    ∑k=1nFk=Fn+2−1\sum_{k=1}^{n} F_k = F_{n+2} - 1

How the math works

Once you know the rule, finding a distant term is mostly substitution. In an arithmetic sequence with a₁ = 2 and d = 3, the fifth term is 2 + (5−1)×3 = 14 because you add 3 four times. The calculator does the same work for whichever n you type. For geometric sequences it uses powers of r (floating-point math), and for Fibonacci it counts up with exact integers so large terms stay accurate up to n = 500. The Logic Trace table shows the intermediate steps so you can follow along rather than trusting a black box. Default arithmetic inputs (a₁ = 2, d = 3, n = 10) produce a₁₀ = 29 and S₁₀ = 155; geometric defaults give a₁₀ = 1,024 and S₁₀ = 2,046; Fibonacci at n = 10 gives F₁₀ = 55 and sum 143.

Controls in Sequence Laboratory

The left side of the widget is where you describe the sequence; the right side is where you read the answers. If any label is unfamiliar, match it to the role below:
  • Sequence type:
    Three toggle buttons pick arithmetic, geometric, or Fibonacci. The input fields change to match.
  • First term (a₁), d, and r:
    a₁ is where the sequence starts. d is the fixed add-on for arithmetic; r is the fixed multiplier for geometric. Fibonacci hides these because the rule is built in.
  • Target n:
    Which term you want (1 through 500). Both the n-th term and Sₙ use this same index.
  • Result, preview bar, Logic Trace, Copy:
    Result holds the headline numbers; First 10 terms (preview) sketches the opening of the list; Logic Trace shows the work; Copy 20 terms to clipboard exports twenty values for reuse elsewhere.

Number Sequence Calculator FAQ

What does the Result panel show?

After you pick a sequence type and enter your numbers, the dark Result panel answers the two questions most homework sets ask for: the value at position n (shown as aₙ or Fₙ) and the running total Sₙ (sum to n) of the first n terms. If you are working in geometric mode and the ratio is between −1 and 1, a third line for the infinite sum may appear as well.

What inputs does each sequence type use?

An arithmetic sequence adds the same amount each step, so you enter the First term (a₁) and the Common difference (d). A geometric sequence multiplies by the same factor each step, so you enter a₁ and the Common ratio (r). The Fibonacci pattern is fixed on this page (F₁ = 1, F₂ = 1, then each term is the sum of the two before it), so you only choose Target n to say which term you want.

What is the Logic Trace table?

The Logic Trace table sits below the copy button and walks through the algebra the way a teacher might on a whiteboard: start with the formula, plug in your values, simplify to get the n-th term, then compute Sₙ. In geometric mode you will also see the sum formula, and when |r| is less than 1, an extra row for S∞. For Fibonacci, the trace lists the early terms and finishes with the neat sum rule F₁ + … + Fₙ = Fₙ₊₂ − 1.

When does the infinite geometric sum appear?

A geometric series can sometimes be added forever and still land on a finite total, but only when the ratio is small enough. On this calculator, switch to Geometric and use a ratio with |r| < 1; the Result panel then shows Sum to Infinity (S∞), which equals a₁/(1−r). If r = 1, every term is the same and the finite sum is simply n·a₁. When |r| is 1 or larger, S∞ is not shown because the series does not converge in that way.

How is Fibonacci indexed on this page?

Different textbooks start Fibonacci at 0 or 1; this tool uses the common convention F₁ = 1 and F₂ = 1, then builds each later term from the two before it. Enter any Target n from 1 to 500 (the widget clamps out-of-range values). Fibonacci values are computed with exact integers, which matters because the numbers grow fast. The sum of the first n terms equals Fₙ₊₂ − 1, and you will see that identity in Logic Trace.

What do the term preview and copy button do?

The First 10 terms (preview) bar shows up to ten chips so you can see how the sequence starts, with +d or ×r labels on arithmetic and geometric modes. Hover a chip to highlight related rows in Logic Trace. Copy 20 terms to clipboard puts the first twenty values in your clipboard as a comma-separated list for spreadsheets or notes; it always exports twenty terms from a₁ or F₁, even when target n is smaller. The helper line under the preview states both limits.

Mathematical Reference Note

Calculation Logic: This tool uses standard mathematical algorithms. While we strive for accuracy, errors in logic or user input can result in incorrect data.

Verification: Results should be cross-checked if used for important academic, professional, or personal calculations.

Standard Terms: This tool is provided free of charge and as-is. CalcRegistry provides no warranty regarding the accuracy or fitness of these results for your specific needs.

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