# Sequence facts for kids

A **sequence** is a word meaning "coming after or next, a series".

It is used in mathematics and other disciplines. In ordinary use it means a series of events, one following another. In maths, a sequence is made up of several things put together, one after the other. The order that the things are in matters: (Blue, Red, Yellow) is a sequence, and (Yellow, Blue, Red) is a sequence, but they are not the same. Sequences made up of numbers are also called **progressions**.

There are two kinds of sequences. One kind is finite sequences, which have an end. For example, (1, 2, 3, 4, 5) is a finite sequence. Sequences can also be infinite, which means they keep going and never end. An example of a sequence that is infinite is the sequence of all even numbers, bigger than 0. This sequence never ends: it starts with 2, 4, 6, and so on, and you can always keep on naming even numbers.

If a sequence is finite, it is easy to say what it is: you can just write down all the things in the sequence. This does not work for an infinite sequence. So another way to write down a sequence is to write a rule for finding the thing in any place you want. The rule should tell us how to get the thing in the *n*-th place, if *n* can be any number. If you know what a function is, this means that a sequence is a kind of function.

For example, the rule could be that the thing in the *n*-th place is the number 2×*n* (2 times n). This tells us what the whole sequence is, even though it never ends. The first number is 2×1, which is 2. The second number is 2×2, or 4. If we want to know the 100-th number, it's 2×100, or 200. No matter which thing in the sequence we want, the rule can tell us what it is.

## Contents

## Types of sequences

### Arithmetic progressions (AP)

The difference between a term and the term before it, is always a constant.

Example:

9 - 4 = 5, 14 - 9 = 5, 19 - 14 = 5, 24 - 19 = 5, and so on

so if you take the first term as A and the constant difference as D the general formula for arithmetic sequence is T=a+(n-1)D where n is the number of term

### Geometric progressions (GP)

The ratio between a term and the term before it, is always constant.

Example:

6 : 3 = 2, 12 : 6 = 2, 24 : 12 = 2, 48 : 24 = 2, and so on

the general formula is T=ar^(n-1) where a is the first term , r is the ratio and n is the number of term.

### Harmonic Progressions (HP)

The difference between the reciprocal of a term and the reciprocal of the term before it, is a constant.

Example:

and so on

## Series

A series is the sum of all the terms of a sequence.

general formula for calculating sum of arithmetic sequence is

S=n/2 [2a=(n-1)d]

that of geometric sequence is

S= a/(1-r) if the sequence is infinite and S= [a(1-r^n)]/(1-r) if it is finite

here a is the first term , d is the common difference in arithmetic sequence , r is the ratio n geometric sequence and n is the number of term.

*Kiddle Encyclopedia.*