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Number Sequence Calculator

Calculators · Added

Arithmetic sequences add a constant; geometric ones multiply by one. Give it a rule and it produces the nth term, the running sum and the sum to infinity where one exists — or give it some terms and it works out which kind you have.

What do you have?
Sequence type

Added to each term

How to use the number sequence calculator

  1. 1Choose whether you have a rule or a list of terms.
  2. 2For a rule: enter the first term, the difference or ratio, and which term you want.
  3. 3Read the nth term, the sum, and the working written out with your numbers in it.
  4. 4For a list: paste at least three terms and it will identify the pattern and continue it.

Examples

An arithmetic sequence

Input
First term 3, difference 5, term 10
Result
a₁₀ = 48, and the first ten terms sum to 255

A converging series

Input
First term 0.9, ratio 0.1
Result
Sum to infinity is exactly 1 — which is the proof that 0.999… = 1

About the number sequence calculator

Where each one turns up

Arithmetic sequences describe anything accumulating at a steady rate: a fixed monthly saving, seats added row by row in a theatre, a countdown. The sum formula — average of the first and last term, times the count — is the one Gauss is said to have found as a schoolboy asked to add the numbers to 100.

Geometric sequences describe anything multiplying: compound interest, population under ideal conditions, radioactive decay, a signal halving through each stage of attenuation. The distinguishing feature is that the ratio, not the difference, is what stays constant — which is why a geometric sequence eventually outgrows any arithmetic one, however fast the arithmetic one is climbing.

0.999… and the converging series

The recurring decimal 0.999… is the geometric series 0.9 + 0.09 + 0.009 + …, with a first term of 0.9 and a ratio of 0.1. Since the ratio is below 1 the series converges, and the sum is 0.9 ÷ (1 − 0.1), which is exactly 1.

That is not an approximation or a convention: the notation 0.999… means the limit of the partial sums, and that limit is 1. The same machinery converts any recurring decimal to a fraction, which is what makes convergence a practical result rather than an abstract one.

Frequently asked questions

When does a geometric series have a sum to infinity?
Only when the absolute value of the ratio is below 1. Then the terms shrink towards zero fast enough for the partial sums to settle, and the total is a₁ ÷ (1 − r). At |r| of 1 or more the terms do not shrink and there is no finite sum — that is a fact about the series rather than a limit of the arithmetic.
Why does the sum formula need a special case?
Because it divides by 1 − r, which is zero when the ratio is 1. That is not an exotic input: the sequence 5, 5, 5, 5 has a ratio of 1 and an obvious sum of 5n. The formula is guarded rather than left to produce a NaN, and it is guarded over a small band rather than at the exact value, because the expression is also numerically poor when the ratio is very close to 1.
How many terms does it need to identify a sequence?
Three at minimum, and three is not many. Two terms fit both patterns and infinitely many others; three fit either pattern exactly. 1, 2, 4 is geometric, and it is also the start of several well-known sequences that are not. The result says how many terms it worked from for that reason.
Why is my sequence neither?
Most real sequences are neither. The Fibonacci numbers add the previous two rather than a constant; the squares have a difference that grows; the primes follow no closed rule at all. Arithmetic and geometric are the two simplest patterns, not a classification of all sequences.

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