Find the nth term and the running sum of the three most common number sequences. Each calculator below works independently.
definition: an = a1 + f × (n−1)
example: 1, 3, 5, 7, 9, 11, 13, ...
definition: an = a × rn−1
example: 1, 2, 4, 8, 16, 32, 64, 128, ...
definition: a0=0; a1=1; an = an−1 + an−2
example: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...
A number sequence calculator finds any term in a sequence without making you write out every term before it, and it computes the running total in the same step. This page covers the three sequences that account for the overwhelming majority of coursework and practical use: arithmetic, geometric, and Fibonacci. Enter the defining values, choose which term you want, and each calculator returns the nth term, the sum of all terms up to it, and a preview of how the sequence begins.
The value of a closed-form calculator becomes obvious the moment the numbers get large. Finding the 50th term of an arithmetic sequence by repeatedly adding is 49 additions and 49 chances to slip; the formula does it in one step. Finding the 100th Fibonacci number by hand is simply not practical - it is a 21-digit number - yet the calculator returns it instantly and exactly.
In mathematics, a sequence is an ordered list of objects. A number sequence, accordingly, is an ordered list of numbers that follow a particular pattern. The individual elements are called terms, and the count of terms is the sequence's length, which may be finite or infinite.
Two properties distinguish a sequence from a set. First, order matters - the sequence 1, 2, 3 is not the same as 3, 2, 1. Second, repetition is allowed - the same value may appear many times, as in 1, 1, 1, 1, or in the Fibonacci sequence where 1 appears twice at the start.
Sequences are usually written with subscript notation: a1 is the first term, a2 the second, and an the general or nth term. A formula for an in terms of n alone is called a closed form or explicit formula, and it lets you jump straight to any term. A formula defining each term from previous ones - as Fibonacci does - is a recursive definition.
An arithmetic sequence adds a fixed amount to get from each term to the next. That fixed amount is the common difference, written f (or often d). The sequence 1, 3, 5, 7, 9, 11, 13, ... is arithmetic with a common difference of 2.
The closed form for the nth term is:
an = a1 + f × (n − 1)
The n − 1 is the part people most often get wrong. The first term needs no additions, the second needs one, the third needs two - so reaching the nth term takes n − 1 steps, not n. Using the calculator's defaults, a first term of 2 with a common difference of 5 gives a 20th term of 2 + 5 × 19 = 97.
The sum of the first n terms, called an arithmetic series, has an elegant closed form:
Sn = n × (a1 + an) ÷ 2
The reasoning is the trick famously attributed to a young Gauss: pair the first term with the last, the second with the second-to-last, and so on. Every pair sums to the same value, a1 + an, and there are n ÷ 2 such pairs. For the default example, S20 = 20 × (2 + 97) ÷ 2 = 990.
The common difference may be negative, producing a decreasing sequence, or a decimal - both work here. A common difference of zero gives a constant sequence, which is technically arithmetic.
A geometric sequence multiplies by a fixed amount to get from each term to the next. That multiplier is the common ratio, written r. The sequence 1, 2, 4, 8, 16, 32, 64, 128, ... is geometric with a common ratio of 2.
The closed form is:
an = a × rn−1
Again the exponent is n − 1, for the same reason: the first term has been multiplied zero times. With a first term of 2 and a ratio of 5, the 12th term is 2 × 511 = 97,656,250 - which shows how quickly geometric growth outpaces arithmetic growth. The arithmetic sequence above reached only 97 by its 20th term.
The sum of the first n terms of a geometric series is:
Sn = a × (rn − 1) ÷ (r − 1) (for r ≠ 1)
When r = 1 every term is identical, so the sum is simply a × n.
Something remarkable happens when the common ratio is between −1 and 1: the terms shrink toward zero fast enough that adding infinitely many of them still gives a finite total. The series is said to converge, and its sum is:
S∞ = a ÷ (1 − r) (only when |r| < 1)
Starting at 1 with a ratio of 0.5 gives 1 + 0.5 + 0.25 + 0.125 + ... which converges to exactly 2, no matter how many terms you add. The calculator shows this extra line automatically whenever |r| < 1. When |r| ≥ 1 the terms do not shrink, the total grows without bound, and the series diverges - so no infinite sum is reported.
This is the mathematics behind the fact that 0.999... equals exactly 1: that repeating decimal is the geometric series 0.9 + 0.09 + 0.009 + ..., with a = 0.9 and r = 0.1, giving 0.9 ÷ 0.9 = 1.
The Fibonacci sequence is defined recursively rather than by a fixed step or ratio. It starts a0 = 0 and a1 = 1, and every term after that is the sum of the two before it:
an = an−1 + an−2
This produces 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... Each number is the sum of its two predecessors: 1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, and so on. The 10th term is 55 and the sum of the first ten terms is 143.
The sum of Fibonacci numbers has a neat identity of its own: the sum of the first n terms equals an+2 − 1. For n = 10, that is a12 − 1 = 144 − 1 = 143, matching the direct addition.
Named after Leonardo of Pisa, known as Fibonacci, who introduced it to Western mathematics in 1202 through a problem about breeding rabbits, the sequence had been described centuries earlier by Indian mathematicians studying Sanskrit poetic metre.
Divide any Fibonacci number by the one before it and the answer creeps toward a constant: 55 ÷ 34 = 1.6176, 89 ÷ 55 = 1.6182, 144 ÷ 89 = 1.6180. That limit is the golden ratio, φ ≈ 1.6180339887, the positive solution of φ² = φ + 1.
This relationship gives a closed form for Fibonacci numbers, known as Binet's formula, expressing an directly in terms of φ without recursion. It is elegant but relies on irrational arithmetic, so for exact results this calculator adds the terms directly instead - which keeps every digit correct even for very large n.
Fibonacci numbers appear widely in nature: the spiral arrangement of seeds in a sunflower head, the count of petals on many flowers, the branching of trees, and the spiral of a nautilus shell all follow Fibonacci-related patterns. The explanation is efficiency rather than mysticism - arranging leaves or seeds at the golden angle packs them most densely and minimises overlap.
Arithmetic sequences grow linearly. Plotted, they form a straight line. Adding a constant each step means the total climbs steadily and predictably. Everyday examples include simple interest, a fixed monthly saving, and seats per row in an auditorium that widens by the same number each row.
Geometric sequences grow exponentially. Plotted, they form a curve that becomes almost vertical. Multiplying each step means growth accelerates dramatically. Examples include compound interest, population growth, radioactive decay (with r < 1), and the doubling of computing power described by Moore's law.
Fibonacci sits between the two. It grows roughly geometrically - each term is about 1.618 times the last - but it is defined by addition rather than multiplication, and it produces whole numbers throughout.
The practical difference is stark. Starting at 2 and stepping by 5, an arithmetic sequence reaches 97 by its 20th term. Starting at 2 with a ratio of 5, a geometric sequence passes 97 million by its 12th. This gap is why compound interest matters so much over long horizons, and why exponential processes are so easy to underestimate.
The term number must be a positive whole number - there is no "term 2.5" or "term −3" in a sequence. Each calculator also shows how its sequence begins, so you can sanity-check that the values you entered describe the pattern you had in mind before trusting the answer.
Finance. Simple interest follows an arithmetic sequence; compound interest follows a geometric one. Loan amortisation, annuity valuation, and depreciation schedules are all sequence problems, and the present value of a perpetuity is an infinite geometric series.
Computer science. Algorithm complexity is described by how work grows with input size - linear growth is arithmetic, exponential growth geometric. Fibonacci numbers appear in Fibonacci heaps and in the standard teaching example of recursion versus memoisation.
Physics and engineering. Radioactive decay is geometric with r < 1, as is the amplitude of a damped oscillation. Signal processing relies heavily on geometric series.
Biology. Unconstrained population growth is geometric; phyllotaxis - the arrangement of leaves and seeds - follows Fibonacci patterns.
Everyday planning. Saving a fixed amount monthly is arithmetic. Doubling a training load each week is geometric. Both are easier to project with a formula than with a spreadsheet full of rows.
Using n instead of n − 1. The single most frequent error in both formulas. The first term has had no differences added and no ratios applied, so the exponent and the multiplier both count from zero.
Confusing the sequence with the series. A sequence is the list of terms; a series is their sum. "The 20th term" and "the sum through the 20th term" are different questions - 97 and 990 respectively in the default example.
Applying the infinite-sum formula when |r| ≥ 1. The formula a ÷ (1 − r) is meaningless for a divergent series. If the ratio is 2, the terms keep growing and the total has no finite value.
Assuming every sequence is arithmetic or geometric. Many are neither. Check by testing both: subtract consecutive terms and see whether the difference is constant, then divide consecutive terms and see whether the ratio is constant. If neither holds, a different rule is at work.
Mixing up index conventions for Fibonacci. Some sources start the sequence at 0 and some at 1. Here the definition sets a0 = 0 and a1 = 1, so the 10th term is 55.
Subtract each term from the one after it. If that difference is the same every time, the sequence is arithmetic. If not, divide each term by the one before it - a constant ratio means it is geometric. The sequence 3, 7, 11, 15 has a constant difference of 4 (arithmetic); 3, 6, 12, 24 has a constant ratio of 2 (geometric).
Yes. A negative common difference gives a decreasing arithmetic sequence. A negative common ratio makes a geometric sequence alternate in sign, as in 1, −2, 4, −8, 16. Both are handled here.
Because it only exists when the common ratio is strictly between −1 and 1. Under that condition the terms shrink toward zero fast enough for the total to converge on a finite value. When |r| ≥ 1 the series diverges and there is no sum to report, so the line is omitted.
354,224,848,179,261,915,075 - a 21-digit number. This calculator computes Fibonacci terms with exact whole-number arithmetic rather than floating-point, so every digit is correct no matter how large n becomes.
For the arithmetic and geometric calculators, the value you enter as "the first number" is term 1, so asking for the 1st term returns exactly what you entered. For Fibonacci, the definition sets a0 = 0 and a1 = 1, so the 1st term is 1 and the 10th is 55.
Yes for the first term and for the common difference or ratio - decimals work throughout, and a ratio of 0.5 is exactly how you model halving. The term number, however, must be a positive whole number, since there is no fractional position in a sequence.
A sequence is the ordered list of terms; a series is what you get by adding them. So 2, 7, 12, 17 is a sequence, and 2 + 7 + 12 + 17 = 38 is the corresponding series. Each calculator reports both the nth term and the series total up to it.
Many. Triangular numbers (1, 3, 6, 10, 15), square numbers (1, 4, 9, 16), prime numbers, and factorials all appear constantly, as do the harmonic series and the Catalan numbers. Arithmetic, geometric, and Fibonacci are simply the three encountered most often, which is why they are the ones covered here.
This Number Sequence Calculator is provided for educational and general informational purposes only. Results for whole-number inputs are computed with exact integer arithmetic; sequences involving decimals are evaluated in standard floating-point arithmetic and displayed to 14 significant digits, so very long or rapidly growing decimal sequences may show minor rounding effects. Verify independently for any critical application.