# Sequences

## Sequences Revision

**Sequences**

A **sequence** is a **list of numbers that follow a pattern**. We can define this pattern with an \color{red}n**th term rule** or a **recurrence relation**. We will also define the terms **increasing sequence**, **decreasing sequence** and **periodic sequence**. We will also define an **arithmetic** and **geometric progression**, both of which will be studied in more detail later on.

**nth Term and Recurrence Relation**

One way to define a **sequence** is with an \color{red}n**th term rule**. This is a **formula** in n, and the **sequence** is generated by n=1, n=2, n=3, etc.

**Example: **4n-3 defines the **sequence** 1,5,9,13... because 4\times 1-3=1, 4\times 2-3=5, 4\times 3-3=9 and so on.

A **recurrence relation** instead **defines the next term using previous ones**. We first define u_{n} as the \color{red}n**th term**, then the **sequence** is **defined by a formula** in terms of u_{n-1}.

**Example: **The **sequence** above, 1,5,9,13... has **recurrence relation** u_{n}=u_{n-1}+4

**Note: **A** recurrence relation** alone is **not enough to specify a sequence**. For example, u_{n}=u_{n-1}+4 defines both 1,5,9,13... and 1000,1004,1008,1012... as well as any other **sequence** that goes up by 4 each time. To **uniquely** define a **sequence**, we **need to specify a term**, e.g. u_{1}=1

**Arithmetic and Geometric Progressions**

An **arithmetic progression** is a** sequence** that is created by **adding the same amount each time**. The **first term** of the** sequence** is called a and the **amount we add** is called the **difference** d.

The **sequence** is a,a+d,a+2d,a+3d...

u_{n}=a+(n-1)d

A **geometric progression** is a **sequence** that is created by **multiplying by the same amount each time**. The **first term** of the **sequence** is called a and the **amount we multiply by** is called r.

The **sequence** is a,ar,ar^{2},ar^{3}...

u_{n}=ar^{n-1}

**Properties of Sequences**

Sequences can be

**Increasing**: u_{n}>u_{n-1}**Decreasing**: u_{n}<u_{n-1}**Periodic**: The sequence**repeats**in a cycle. The number of terms in the cycle is the order of the sequence.

Some sequences are **none of these**.

**Example 1: Arithmetic and Geometric Progressions**

a) 3,7,11,15,19 is an **arithmetic progression**. Find a and d, and **predict the next term**.

b) 6,30,150,750,3750 is a **geometric progression**. Find a and r, and **predict the next term**.

**[4 marks]**

a) a=\text{first term}=3

d=\text{difference}=7-3=4

\begin{aligned}u_{n}&=3+4(n-1) \\[1.2em] u_{6}&=3+4(6-1)\\[1.2em]&=3+4\times 5\\[1.2em]&=3+20\\[1.2em]&=23\end{aligned}

b) a=\text{first term}=6

r=\text{ratio}=\dfrac{30}{6}=5

\begin{aligned}u_{n}&=6\times 5^{n-1}\\[1.2em] u_{6}&=6\times 5^{6-1}\\[1.2em]&=6\times 5^{5}\\[1.2em]&=6\times 3125\\[1.2em]&=18750\end{aligned}

**Example 2: Properties of Sequences**

Prove that the **sequence** defined by 8n-7 is **increasing**.

**[2 marks]**

u_{n}=8n-7

u_{n-1}=8(n-1)-7

\begin{aligned}u_{n}-u_{n-1}&=8n-7-(8(n-1)-7)\\[1.2em]&=8n-7-(8n-8-7)\\[1.2em]&=8n-7-(8n-15)\\[1.2em]&=8n-7-8n+15\\[1.2em]&=8>0\end{aligned}

Hence, u_{n}>u_{n-1} so the sequence is increasing.

## Sequences Example Questions

**Question 1: **Express the following sequence in both nth term and recurrence relation form.

**[2 marks]**

nth term: u_{n}=3n-2

Recurrence relation: u_{n}=u_{n-1}+3 and u_{1}=1

**Question 2: **

i) An arithmetic series has first term 64 and difference 1.5. Find the seventh term in the sequence.

ii) A geometric series has first term 64 and ratio 1.5. Find the seventh term in the sequence.

**[4 marks]**

i) u_{n}=64+1.5(n-1)

\begin{aligned}u_{7}&=64+1.5(7-1)\\[1.2em]&=64+1.5\times 6\\[1.2em]&=64+9\\[1.2em]&=73\end{aligned}

ii) u_{n}=64\times 1.5^{n-1}

\begin{aligned}u_{7}&=64\times 1.5^{7-1}\\[1.2em]&=64\times 1.5^{6}\\[1.2em]&=64\times 11.390625\\[1.2em]&=729\end{aligned}

**Question 3: **Below are seven sequences:

A: 2,5,8,11,14,17,20,23

B: 2,4,7,8,14,15,19

C: 31,28,31,30,31,30,31,31,30,31,30,31

D: 99,98.5,98.25,98.125,98.0625

E: 10000,9000,8100,7290,6561

F: 1,0,-1,0,1,0,-1,0,1,0,-1,0

G: 24,48,96,192,384,768,1536

Which of these sequences are:

i) Increasing

ii) Decreasing

iii) Periodic

iv) Arithmetic progression

v) Geometric progression

vi) None of the above

**[4 marks]**

i) A, B, G

ii) D, E

iii) F

iv) A

v) E, G

vi) C