The terms sequence and series sound very similar, but they are quite different. Basically:

- A
**sequence**is a set of ordered numbers, like 1, 2, 3, …, - A
**series**is the*sum*of a set of numbers, like 1 + 2 + 3….

**Sequence and Series: Contents (Click to skip to that section):**

## What is a Sequence?

A **sequence **is a collection of elements (usually numbers) with two major differences from plain old “sets”:

- The elements (or terms) are in order.
- The values of the terms can repeat.

The number of elements is called the *length *of the sequence. The length is potentially infinite, meaning it goes on without end. If the collection of numbers ends then it is called a **finite sequence**.

## Sequence Rules and Arithmetic Sequences

Sequences generally have a **rule**. The rule is applied to find the value of unknown terms. For example, the sequence {3, 6, 9, 12…} begins at 3 and increases by 3 for every subsequent value. This is an example of an **arithmetic sequence** with a **common difference** of 3.

The above rule is helpful in defining values in the immediate vicinity of the preceding terms. A limitation of the rule, however, is that it does not tell us the value of an unknown term. In the example above, we don’t know what the 100*th* term will be. In other words, we need a rule that can quickly determine the value for any unknown term.

## Formula and Example for Nth Term

The formula for the *n*th term of the arithmetic sequence is:

**A _{n} = a + (n -1)d**

Where:

- a
_{n}is the nth term of the sequence, - a is the first term,
- d is the common difference.

The *n*th term is the unknown term that you are trying to calculate. Using the formula above you can quickly calculate any *n*th term.

**Example question**: Find the 100th term for {3, 6, 9, 12…}:

Step 1: Determine the values for *a*_{n}, *d* and *n*.

- a
_{n}(the nth term that you are trying to find) = 100, - a is the first term: 3,
- d is the common difference (3),

Step 2: Place each value from Step 1 into the formula:

*a*_{100} = 3 + (100 – 1) 3

= 3 + (99) 3

= 3 + 297

*a*_{100} = 300

The 100th term is 300.

## Types of Sequences

Arithmetic Progression

Bounded Sequence: Definition

Fibonacci Sequence

Finite Geometric Sequence

Infinite Sequence

Infinite Geometric Sequence

Monotonic Sequence / Series

n-tuple: Simple Definition, Examples

Orthogonal Polynomials

Sequence of Partial Sums

## What is a Series?

A series is what you get when you add up the terms in a sequence; when you add these terms, you’re performing what’s more formally called a summation (which is just a fancy way of saying “add them all up”).

## Types of Series

Alternating Series

Alternating Harmonic Series

Binomial Series

Continuous Series & Discrete Series

Divergent Series

Finite Series

Geometric Series

Harmonic Series

Infinite Arithmetic Series

Infinite Series

Laurent Series

Monotonic Sequence / Series

Oscillating Series

P-Series

Positive Series: Definition, Examples, Convergence

Power Series

Telescoping Series

## Sequence and Series: Related Articles

Alternating Series Test

Pointwise Convergence

Series Rules

Remainder of a Series: Step by Step Example, How to Find

Sum of a Convergent Geometric Series

Uniform Convergence

## References

Larson, R. & Edwards, B. (2013). Calculus. Cengage Learning; 10 edition

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