# Sequences

A **sequence** is an ordered list of numbers or objects arranged according to a rule.

A sequence of numbers is a function s: **N** → **R**

Each number of the sequence is called a **term**, and we represent it as a_{i}. We call **general term** of a sequence, a_{n}, the expression which represents any term of the sequence.

For example:

· 1, 2, 3, 4, 5, … a_{n} = n, then a_{100} = 100

· 2, 4, 6, 8, 10, … a_{n} = 2n, then a_{25} = 50

· 1, 4, 9, 16, 25, … a_{n} = n^{2}, then a_{12} = 144

· 32, 16, 8, 4, 2, … a_{n} = 2^{6-n}, then a_{10} = 2^{-4} = 1/16

Some sequences are recurring, because we obtain each term from the previous one. For example:

· 1, 1, 2, 3, 5, 8, 13, … a_{n} = a_{n-1} + a_{n-2} a_{1} = a_{2} = 1

This is the Fibonacci sequence.

An **arithmetic progression** is such a sequence of numbers that the difference of any two successive members of the sequence is a constant. The constant is called **difference** of the progression.

Example: 3, 5, 7, 9, …. d = 2

General term: a_{n} = a_{1} + (n-1)· d

In the example: a_{n} = 3 + (n-1)· 2 = 2n + 1

A **geometric progression** is such a sequence of numbers that the quotient of any two successive members of the sequence is a constant, which is called **ratio** of the progression.

Example: 3, 6, 12, 24 …. r = 2

General term: a_{n} = a1· r^{n-1}

In the example: a_{n} = 3 · 2^{n-1}

**Exercise**: find the general term and a_{10} of these sequences:

a) 1, 3, 9, 27, 81, ...

b) 15, 12, 9, 6, 3, 0, ...

c) 1, -2, 3, -4, 5, ...

Solutions: a) a_{n}=3^{n-1}, a_{10}= 3^{9}= 19683; b) a_{n}= 18- 3n, a_{10}=-12; c) (-1)^{n}·n; a_{10}= 10;

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