### Sequences

#### Introduction to Sequences

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# Learning Objectives

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###### What are sequences?

Sequences are sets of items belonging in a specific order, according to a certain rule.

Each member of a sequence is referred to as a term.

An example of a simple sequence is 5, 10, 15, 20, 25.

Consider the terms of the above sequence. The first term is 5. We write this as T₁=5. The second term is 10, which is written as T₂=10. And, since the third term is 15, T₃=15.

###### Sequence notation

When dealing with sequences, we use a special notation. The above table summarises this.

###### Types of sequences

We will be required to work with two main types of sequences:** arithmetic sequences **(also known as APs) and **geometric sequences **(or GPs).

An **Arithmetic Progression (AP)** is some sequence of numbers in which each term is obtained from the previous term by the **addition **of some constant number.

A** Geometric Progression (GP)** is one where the next term in the sequence is obtained from **multiplying **the previous term by a constant number.

###### Arithmetic sequences / progressions (APs)

Sequences of numbers in which each term is obtained from the previous term by the addition of some constant number are said to be arithmetic sequences or APs.

**Thus, all arithmetic sequences are of the form:**

** **

𝒂, 𝑎+𝒅, 𝑎+2𝑑, 𝑎+3𝑑, 𝑎+4𝑑, 𝑎+5𝑑, 𝑎+6𝑑.

In this general form, we have:

A first term of 𝒂

A common difference of 𝒅.

Therefore, Tₙ₊₁ = Tₙ + d.

###### Geometric sequences / progressions (GPs)

All GPs are of the form **𝒂, 𝑎𝒓², 𝑎𝑟³, 𝑎𝑟⁴, 𝑎𝑟⁵, 𝑎𝑟⁶, 𝑎𝑟⁷.**

In this general form, we have:

A first term of

**a**; andA

**common ratio**of**r**.

Using **recursive notation**, we write this as Tₙ₊₁ = 𝑟×Tₙ with T₁=𝑎.