# Sequences

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## 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.

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## 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. $u_{1}=1$

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## 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 $a,ar,ar^{2},ar^{3}...$

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

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## Properties of Sequences

Sequences can be

• Increasing: $u_{n}>u_{n-1}$
• Decreasing: $u_{n}
• 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.

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## 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}

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## 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.

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## Sequences Example Questions

Question 1: Express the following sequence in both $n$th term and recurrence relation form.

$1,4,7,10,13,16,19,22$

[2 marks]

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$n$th term: $u_{n}=3n-2$

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

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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]

A Level AQAEdexcelOCR

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}

Gold Standard Education

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]

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i) A, B, G

ii) D, E

iii) F

iv) A

v) E, G

vi) C

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A Level

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