Sequences

A sequence is just a set of numbers that follows a specific rule. This rule determines how each term in the sequence relates to the previous one. The rule could be simple, like adding a constant number each time, or more complex, involving squares or other operation

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Things You Need to Be Able to Do with Sequences:

1. Spotting and Describing Number Sequences

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Example Sequences: Let's look at some common types of sequences:

Linear Sequence (Addition/Subtraction) Sequence: $7,10,13,16,19,…$

Rule: The numbers are increasing by $3$ each time.

This is an arithmetic sequence where each term is found by adding a fixed number to the previous term.

Predicting Next Numbers:

Add $3$ to the last number:
$19+3=22$
$22+3=25$
Next two numbers: $22,25$

Geometric Sequence (Multiplication/Division) Sequence: $3,6,12,24,48,…$

Rule: The numbers are doubling each time.

This is a geometric sequence where each term is found by multiplying the previous term by a fixed number.

Predicting Next Numbers:

Multiply the last number by $2$ :
$48×2=96$
$96×2=192$
Next two numbers: $96,192$

Complex Arithmetic Sequence (Varying Differences) Sequence: $200,190,181,173,166,…$

Rule: The numbers are decreasing with a pattern. The differences between the terms are decreasing by $1$ each time.

Subtract $10$ , then $9$ , then $8$ , and so on. Predicting Next Numbers:
Subtract the next difference ( $7$ , then $6$ ):
$166−7=160$
$160−6=154$
Next two numbers: $160,154$

Fibonacci Sequence Sequence: $1,1,2,3,5,8,13,…$

Rule: Each number is the sum of the two previous numbers.

This sequence is known as the Fibonacci sequence.

Predicting Next Numbers:

Add the last two numbers together:
$8+13=21$
$13+21=34$
Next two numbers: $21,34$

Worked Example: Exam-Style Question

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Question: The sequence $5,11,23,47,…$ follows a pattern. Describe the rule and find the next two terms.

Step-by-Step Solution:

Identify the Pattern:

Look at the differences between terms:
$11−5=6$
$23−11=12$
$47−23=24$
The differences are doubling each time ( $6, 12, 24$ ).

Describe the Rule:

The difference between the terms doubles each time.

Predict the Next Numbers:

The next difference should be $24×2=48$ .
Add $48$ to the last number:
$47+48=95$
The next difference should be $48×2=96$
Add $96$ to $95$ :
$95+96=191$
Next two numbers: $95,191$

Final Answer: The next two terms are $95$ and $191$ .

2. Finding the $nth$ Term of Linear Sequences

What is the $nth$ Term?

The $nth$ term is a formula that represents the general term of a sequence, where $n$ represents the position of the term in the sequence. For example, $n=1$ for the first term, $n=2$ for the second term, and so on.

In linear sequences, the $nth$ term can be written in the form:

nth term=an+b

Where:

$a$ is the common difference (what you add or subtract each time).
$b$ is a constant that adjusts the sequence to match the terms.

Steps to Find the $nth$ Term of a Linear Sequence

Determine the Common Difference ( $a$ )
Write the Times Table of $a$
Calculate the Constant ( $b$ )

Determine the Common Difference ( $a$ ):

Identify the amount you add or subtract to get from one term to the next. This value is your $a$ .

Write the Times Table of $a$ :

List out the multiples of $a$ . This will help you figure out the base sequence that your sequence is related to.

Calculate the Constant ( $b$ ):

Compare the sequence you are given with the multiples of $a$ to figure out what needs to be added or subtracted to get the sequence you're working with.

Worked Examples:

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Example 1: Find the $nth$ term of the sequence $13, 19, 25, 31, 37, ...$

Identify the Common Difference:

The difference between the terms is $+6$ . So, $a=6$ .

Write the $6$ Times Table:

List the first few multiples of $6$ :

6,12,18,24,30,36,…

Compare to the Given Sequence:

The given sequence is:

13,19,25,31,37,…

Notice that each term in the sequence is $7$ more than the corresponding term in the $6$ times table.

Write the $nth$ Term Formula:

The $nth$ term is $6n+7$ .

Test the Formula:

To find the $5th$ term $(n=5)$ :

6(5)+7=30+7=37

The formula works, so the $nth$ term is $6n+7$ .

Final Answer: The $nth$ term of the sequence is $6n+7$ .

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Example 2: Find the $nth$ term of the sequence $−20,−16,−12,−8,−4,…$

Step 1: Identify the Common Difference

Look at the sequence and see how much you add or subtract to move from one term to the next.
Here, you add $4$ each time.
$−16−(−20)=4$
$−12−(−16)=4$
$−8−(−12)=4$
So, the common difference $d$ is $+4$ .

Step 2: Write the Multiples of the Common Difference

Since the common difference is $4$ , write down the $4$ times table under the sequence to help identify the $nth$ term:

Sequence:−20,−16,−12,−8,−4,…

4n:4,8,12,16,20,…

Step 3: Determine the Constant Term ( $b$ )

Compare the given sequence with the $4$ times table to see what you need to subtract or add to get from the multiples of $4$ to the actual sequence.
To go from the $4$ times table to the actual sequence, you subtract $24$ from each term.
$4−24=−20$
$8−24=−16$
$12−24=−12$
$16−24=−8$
$20−24=−4$
So, the constant term $b$ is $−24$ .

Step 4: Write the nth Term Formula

Now that you have the common difference and the constant term, you can write the $nth$ term formula:

nth term=4n−24

Step 5: Test the Formula

Use the formula to find a specific term in the sequence to ensure it's correct.
For example, to find the $5th$ term $(n=5):$

4(5)−24=20−24=−4

The $5th$ term is $−4$ , which matches the sequence.

Final Answer: The $nth$ term of the sequence is $4n−24$ .

Predicting a Specific Term Using the $nth$ Term Formula:

You can use the $nth$ term formula to predict any term in the sequence without writing out the entire sequence.

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Example: Find the $100th$ term of the sequence. 13. Substitute $n=100$ into the formula:

100th term=4(100)−24=400−24=376

Final Answer: The $100th$ term of the sequence is $376$ .

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Example 3: Find the $nth$ term of the Sequence: $21,16,11,6,1,…$

Step 1: Identify the Common Difference

Observe how much you subtract each time to move from one term to the next.
Here, you subtract $5$ each time:
$16−21=−5$
$11−16=−5$
$6−11=−5$
$1−6=−5$
So, the common difference $d$ is $−5$ .

Step 2: Write the Multiples of the Common Difference

Since the common difference is $−5$ , write down the multiples of $−5$ under the sequence to help identify the $nth$ term:

Sequence: 21,16,11,6,1,…

5n:−5,−10,−15,−20,−25,…

Step 3: Determine the Constant Term ( $b$ )

Compare the given sequence with the multiples of $−5$ to see what you need to add or subtract to get from the multiples of $−5$ to the actual sequence.
To go from the $-$$5$ times table to the actual sequence, you add $26$ to each term.
$−5+26=21$
$−10+26=16$
$−15+26=11$
$−20+26=6$
$−25+26=1$
So, the constant term $b$ is $+26$ .

Step 4: Write the $nth$ Term Formula

Now that you have the common difference and the constant term, you can write the $nth$ term formula:

nth term=−5n+26

Step 5: Test the Formula

Use the formula to find a specific term in the sequence to ensure it's correct.
For example, to find the $6th$ term ( $n=6$ ):

−5(6)+26=−30+26=−4

The $6th$ term is $−4$ , which would be the next term in the sequence if it continued.

Final Answer: The $nth$ term of the sequence is $−5n+26$

3. Writing Out the Terms of a Sequence Given the Rule

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Example 1: Writing Out the First $5$ Terms Given $nth$ term rule:

nth term=7n−3

Step-by-Step Solution:

$1st$ Term $(n = 1)$ :

1st term=7(1)−3=7−3=4

$2nd$ Term $(n = 2)$ :

2nd term=7(2)−3=14−3=11

$3rd$ Term $(n = 3)$ :

3rd term=7(3)−3=21−3=18

4. $4th$ Term $(n = 4)$ :

4th term=7(4)−3=28−3=25

5. $5th$ Term $(n = 5)$ :

5th term=7(5)−3=35−3=32

Result: The first $5$ terms of the sequence are $4,11,18,25,32$

Note: The difference between each term is $7$ , which matches the coefficient of $n$ in the $nth$ term formula.

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Example 2: Writing Out the First $5$ Terms Given $nth$ term rule:

nth term=n²+10

Step-by-Step Solution:

$1st$ Term $(n = 1)$ :

1st term=1²+10=1+10=11

$2nd$ Term $(n = 2)$ :

2nd term=2²+10=4+10=14

$3rd$ Term $(n = 3)$ :

3rd term=3²+10=9+10=19

$4th$ Term $(n = 4)$ :

4th term=4²+10=16+10=26

$5th$ Term $(n = 5)$ :

5th term=5²+10=25+10=35

Result: The first $5$ terms of the sequence are $11,14,19,26,35$

4. Working Out the $n$ th Term of Quadratic Sequences

In a quadratic sequence, the difference between consecutive terms changes, but the second difference (the difference of the differences) is constant. The nth term of a quadratic sequence generally takes the form:

\text{nth term} = an^2 + bn + c

However, a simpler method often works when you can recognise square numbers.

Step-by-Step Method

Step 1: Identify Square Numbers

Step 2: Determine the Rule

Step 1: Identify Square Numbers

Write out the square numbers ( $n^2$ ) underneath your sequence.
This helps you spot how your sequence relates to a basic sequence of squares.

Step 2: Determine the Rule

Compare the sequence with the square numbers.
Determine what you need to add or subtract from the square numbers to get your sequence.

Worked Example:

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Example Sequence: $−2,1,6,13,22,…$

Step 1: Write Out the Square Numbers

List the first few square numbers ( $n^2$ ) underneath the sequence:

n:1,2,3,4,5,…

Sequence:−2,1,6,13,22,…

n^2:1,4,9,16,25,…

Step 2: Determine What to Add or Subtract

Compare each term in the sequence with the corresponding square number:
$−2=1−3$
$1=4−3$
$6=9−3$
$13=16−3$
$22=25−3$
You consistently subtract $3$ from the square number to get the sequence term.

Final $n$ th Term Formula

Since you subtract $3$ from each square number:

nth term=n2−3

Final Answer: The $nth$ term of the sequence is $n2−3$

Sequences Simplified Revision Notes for GCSE OCR Maths

Sequences

1. Spotting and Describing Number Sequences

Worked Example: Exam-Style Question

2. Finding the $nth$ Term of Linear Sequences

Steps to Find the $nth$ Term of a Linear Sequence

Worked Examples:

Example 1: Find the $nth$ term of the sequence $13, 19, 25, 31, 37, ...$

Example 2: Find the $nth$ term of the sequence $−20,−16,−12,−8,−4,…$

Predicting a Specific Term Using the $nth$ Term Formula:

Example 3: Find the $nth$ term of the Sequence: $21,16,11,6,1,…$

3. Writing Out the Terms of a Sequence Given the Rule

4. Working Out the $n$ th Term of Quadratic Sequences

Step-by-Step Method

Worked Example:

Example Sequence: $−2,1,6,13,22,…$

500K+ Students Use These Powerful Tools to Master Sequences For their GCSE Exams.

Other Revision Notes related to Sequences you should explore

Sequences Simplified Revision Notes for GCSE OCR Maths

Sequences

1. Spotting and Describing Number Sequences

Worked Example: Exam-Style Question

2. Finding the nthnth nth Term of Linear Sequences

Steps to Find the nthnth nth Term of a Linear Sequence

Worked Examples:

Example 1: Find the nthnth nth term of the sequence 13,19,25,31,37,...13, 19, 25, 31, 37, ...13,19,25,31,37,...

Example 2: Find the nthnthnth term of the sequence −20,−16,−12,−8,−4,…−20,−16,−12,−8,−4,…−20,−16,−12,−8,−4,…

Predicting a Specific Term Using the nthnthnth Term Formula:

Example 3: Find the nthnthnth term of the Sequence: 21,16,11,6,1,…21,16,11,6,1,…21,16,11,6,1,…

3. Writing Out the Terms of a Sequence Given the Rule

4. Working Out the nnnth Term of Quadratic Sequences

Step-by-Step Method

Worked Example:

Example Sequence:−2,1,6,13,22,… −2,1,6,13,22,…−2,1,6,13,22,…

500K+ Students Use These Powerful Tools to Master Sequences For their GCSE Exams.

Other Revision Notes related to Sequences you should explore

2. Finding the $nth$ Term of Linear Sequences

Steps to Find the $nth$ Term of a Linear Sequence

Example 1: Find the $nth$ term of the sequence $13, 19, 25, 31, 37, ...$

Example 2: Find the $nth$ term of the sequence $−20,−16,−12,−8,−4,…$

Predicting a Specific Term Using the $nth$ Term Formula:

Example 3: Find the $nth$ term of the Sequence: $21,16,11,6,1,…$

4. Working Out the $n$ th Term of Quadratic Sequences

Example Sequence: $−2,1,6,13,22,…$