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Math & Statisticssequences

Arithmetic & Geometric Progression Calculator

Arithmetic and geometric progressions: the nth term and the running sum. The difference between adding and multiplying each step is the difference between linear and exponential, which is most of what the topic is for.

Also called: arithmetic progression, geometric progression, sequence calculator.

Sequence type
The nth term
14

Term 5 is 14 and the first 5 terms sum to 40. The sequence starts 2, 5, 8, 11, 14.

Sum of the first n terms
40
First terms
2, 5, 8, 11, 14
About this sequence
Adding a fixed amount each step, so the terms lie on a straight line.
How this is calculated
Method and background

How this is calculated

An arithmetic progression adds the same amount each step, so its terms lie on a straight line and its sum is the number of terms times their average. A geometric progression multiplies by the same factor, so its terms curve away and its sum comes from the ratio formula. A ratio of exactly one is the degenerate case where the formula divides by zero and the answer is simply the term repeated.

arithmetic adds a fixed difference each time; geometric multiplies by a fixed ratio
a
First term
d
Common difference
r
Common ratio
n
Number of terms

Worked examples

Each of these is asserted on every build. If a change to the engine ever moved one of these answers, the build would fail before the page could print it.

an arithmetic progression

Sequence type
Arithmetic (add each time)
First term
2
Common difference or ratio
3
Number of terms
5

The nth term14

2, 5, 8, 11, 14 summed by hand

Open this example

a geometric progression

Sequence type
Geometric (multiply each time)
First term
3
Common difference or ratio
2
Number of terms
5

The nth term48

3, 6, 12, 24, 48 summed by hand

Open this example

a ratio of one repeats the term

Sequence type
Geometric (multiply each time)
First term
3
Common difference or ratio
1
Number of terms
4

The nth term3

boundary: the case where the standard formula divides by zero

Open this example

Method and limits

What it assumes

  • A finite number of terms.

What it deliberately does not model

  • The preview lists at most five hundred terms even when more are summed.
  • An infinite geometric series converges only when the ratio is between minus one and one, which this does not compute.

Formula version 1.0.0 · definition 1.0.0 · United States · Report a problem with this calculator

Frequently asked questions

What happens when the ratio is one?
Every term is the first term, so the sum is simply that term times the count. The standard formula divides by ratio minus one and would fail, so it is handled separately.