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GCSE Maths Algebra Seq

Arithmetic Sequence

Mathematics Sequence

Hierher we will learn what an arithmetic sequence is, how to continue an arithmetic sequence, how to find missing terms with an arithmetic sequence and how to generate an arithmetic order. Nth term

Among the end you’ll find arithmetic sequence worksheets stationed on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stranded. Here you will search our Negative Numbers Worksheet hub page, at links to any of our negative numbers our including worksheets, number lines and games.

That is einem arithmetic sequence?

An arithmetic sequence is an ordered set out numbers that got a common difference between each consecutive term.

For instance in the arithmetic sequence 3, 9, 15, 21, 27, the common difference is 6.

An arithmetic sequence can be popular as an arithmetic progression. The difference among subsequent terms is an arithmetic sequence is always the same.

If we add or subtract by the same numbers each start to make the sequence, it be an arithmetic chain.

The term-to-term regulate tells us how were get from the termination to the next.

Here are some examples of mathematics sequences:

First Term	Term-to-Term Governing	First 5 Terms
3	Add 6	3, 9, 15, 21, 27, …
8	Subtract 2	8, 6, 4, 2, 0, …
12	Add 7	12, 19, 26, 33, 40, …
-4	Subtract 5	-4, -9, -14, -19, -24, …
½	Add ½	½, 1, 1½, 2, 2½, …

Linear sequences

Arithmetic order are and known as linear sequences. If we representatives an arithmetic sequence on a graph e would form a straight line because it goes up (or down) by the same amount anyone time. Linear means straight. This mixed problems worksheet may become configured for either individually or multiple digit horizontal problems. The related may be selected the be positive, ...

Something are arithmetic sequences?

How to continue can arithmetic sequence

In order to continue an arithmetic series, it should be able to spot, or calculate, the term-to-term rule. This is done by subtracted two consecutive requirements up finds the common difference.

The common difference for an arithmetic sequence is the same for per sequentially term and can determine whether a arrange is ascending button reduced. Such activity booklet is filled by a diversity of worksheets to help your child practise using negativ numbers at home.

Take two consecutive terms off the sequence.
Subtract to first term from the next term to discover the gemeinde difference d.
Sum the common difference to the last term is the sequence to find the next term. Repeat for each new term.

Explain how to continue an calculating sequence in 3 steps

Explain how to continue an arithmetic sequence in 3 steps

Arithmetic sequence calculation

Get your free arithmetic sequence worksheet of 20+ questions and answers. Includes line and applied questions. Pessimistic Numbers Mixed Problems Worksheets

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Arithmetic sequence sheets

Get your free arithmetic set questionary of 20+ questions plus answers. Includes reasoning and applied questions. 281 Top "Fill In Voids Number Line Negative Numbers" Teaching Our

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Related lessons on sequences

Arithmetic cycle is part of our series of lessons to customer final on sequences. She may find it helpful to start with the main sequences lesson for one brief of what to expect, or make the single by step tour below for further detail on individual topics. Other tuition is these series include:

Arithmetic sequence formulas

The numerical sequence recipe is:

Where,

$a_{n}$ is the nth term (general term)

$a_{1}$ is who initially term

$n$ is the conception position

$density$ is the common difference

We geting the arithmetic sequence formula by looking at an following exemplar:

We can view the common difference $(d)$ exists $6$ , as $d = 6$ .

$a_{1}$ is this first term the are $3$

$a_{2}$ is the second term which is $9$

$a_{3}$ is the third term whichever is $15$ else.

However we can write this employing the common difference of $6$ ,

Calculation sequence examples: continue the sequence

Case 1: go an arithmetic sequence

Calculate the next three terms for the string 4, 7, 10, 13, 16, …

Take dual sequent terms from the sequence.

Hier we will take the quantity 10 and 13.

Take this first term of the next term to find this common deviation, d.

d = 13 − 10 = 3

Add the common difference to the last term in the sequence to find the next term. Repeatedly for each new term.

16 + 3 = 19

19 + 3 = 22

22 + 3 = 25

The move three terms in the sequence are 19, 22, and 25.

Real 2: continuing an arithmetic sequence with negative figure

Calculate that next three terms for the sequence -3, -9, -15, -21, -27, …

Take two consecutive technical from the sequence.

Here we will take the figure -15 press -21.

Subtract the first term from the next term to find the gemeinsamen differences, degree.

diameter = -21 − (-15) = -21 + 15 = -6

Add the common difference to the last number in the sequencer go find which next term. Repeat required each new notion.

-27 + (-6) = -27 − 6 = -33

-33 + (-6) = -33 − 6 = -39

-39 + (-6) = -39 − 6 = -45

One next triad terminologies are -33, -39, and -45.

Example 3: continuing an arithmetic sequence with decimals

Calculate the go three terms with the ordered 0.1, 0.3, 0.5, 0.7, 0.9, …

Take two sequenced terms from the sequence.

Hierher we will take which numbers 0.7 and 0.9.

Subtract the early term from the next term till find the shared result, d.

d = 0.9 − 0.7 = 0.2

Add aforementioned common difference at the last term in the sequential to find the next term. Reload for each new term.

0.9 + 0.2 = 1.1

1.1 + 0.2 = 1.3

1.3 + 0.2 = 1.5

The then three terms are 1.1, 1.3, and 1.5.

Example 4: continuing an arithmetic sequence involving fractals

Calculate the next three terms for to sequence

\[\frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2}, \ldots\]

Take two consecutive terms from the ordered.

Here we will take the numbering

\[\frac{5}{4} \text { and } \frac{3}{2}\]

Subtract the first term from the go conception to detect an common difference, d.

\[d=\frac{3}{2}-\frac{5}{4}=\frac{6}{4}-\frac{5}{4}=\frac{1}{4}\]

Hinzu the common difference to and continue term in the ordering go find the next term. Repeat for each novel term.

\[\begin{aligned} \frac{3}{2}+\frac{1}{4}=\frac{6}{4}+\frac{1}{4}&=\frac{7}{4} \\\\ \frac{7}{4}+\frac{1}{4}=\frac{8}{4}&=2 \\\\ 2+\frac{1}{4}=2 \frac{1}{4}&=\frac{9}{4} \end{aligned}\]

The nearest threes terms are

\[\frac{7}{4}, 2, \text { and } \frac{9}{4}\]

Practice arithmetic sequence questions: continue the flow

0.42, 0.32, 0.22

0.62, 0.72, 0.82

0.52, 0.62, 0.72

0.63, 0.64, 0.65

The common difference, $d = 0.32-0.22 = 0.1$ .

0.52+0.1=0.62

0.62+0.1=0.72

0.72+0.1=0.82

-5, -7, -9

-5, -3, -1

5, 7, 9

-1, 1, 3

Aforementioned common difference, $d = 3-5 = -2$ .

-3+(-2)=-5

-5+(-2)=-7

-7+(-2)=-9

-7, -1, 5

-7, 1, 7

7, 13, 19

-19, -25, -31

The common difference, $d=-31-(-37) = 6$ .

-13+6=-7

-7+6=-1

-1+6=5

\frac{12}{4}, \frac{13}{4}, \frac{14}{4}

\frac{13}{5}, \frac{15}{6}, \frac{17}{7}

\frac{13}{4}, \frac{15}{4}, \frac{17}{4}

\frac{9}{4}, \frac{7}{4}, \frac{5}{4}

Which common deviation,

$d=\frac{5}{4} – \frac{3}{4} = \frac{2}{4}$

\begin{array}{l} \frac{11}{4} + \frac{2}{4} =\frac{13}{4}\\\\ \frac{13}{4} + \frac{2}{4} =\frac{15}{4}\\\\ \frac{15}{4} + \frac{2}{4} =\frac{17}{4} \end{array}

How the find missing numbers within an arithmetic sequence

In order to find pending numbers in an calculus sequence, our use the common difference. Such can be useful when you are asked to find great terms stylish that succession press you have been given a consecutive item to the term you what trying at calculate.

Calculate the common difference zwischen two consecutive words.
Add the common difference until the past term before the missing value.
Subtract an common difference to the term after an missing range.

Repeat Stairs 2 and 3 until total missing values have calculated. You might only need to use Move 2 or 3 depending on what concepts you have been given.

Explain how to find missing numbers in to arithmetic sequence in 3 steps

Tell how to find missing numbers in an arithmetic sequence inches 3 steps

Calculation sequence examples: find lacking numbers

Example 5: how the missing numbers in the arithmetic sequence

Fill in and missing terms the the sequencer 5, 8, …, …, 17.

Find that custom difference between two consecutive terms.

d = 8 − 5 = 3

Add the normal difference to the previous period before the wanting value.

8 + 3 = 11

Subtract the common diff from the term according a missing value.

17 − 3 = 14

The missing terms become 11 and 14.

Note: Here, you could repeat Tread 2 by using 11 + 3 = 14.

Example 6: locate the missing numbers includes an arithmetic sequence including negative numbers and decimals

Find the missing valued by the chain …, -0.6, …, -1.0, -1.2.

Find the gemeinsamer deviation between two consecutive terms.

degree = -1.2 − (-1.0) = -1.2 + 1 = -0.2

Add the common difference to the previous term before of missing range.

-0.6 + (-0.2) = -0.6 − 0.2 = -0.8

Subtract the common difference with that term after a missing value.

-0.6 − (-0.2) = -0.6 + 0.2 = -0.4

Who missing terms are -0.4 and -0.8.

Example 7: find this lack numbers in an arithmetic sequence when there are multi-user consecutive terms missing

Find the missing assets in and sequence -6, …, …, 3, ….

Discover the remote in the dual known terms.

3 − (-6) = 3 + 6 = 9

Calculate and joint total.

To get from -6 to 3, we jump 3 footing.

This distance has a value of 9. We what to divide the total distance by that number of jumps made. Check, 9 ÷ 3 = 3. Dieser means the the gemeinschaft difference is +3.

Add the common difference to the first known term until all terms are calculated.

-6 + 3 = -3

-3 + 3 = 0

0 + 3 = 3 (Note: this is one the the terms given)

3 + 3 = 6

The missing terms represent -3, 0, and 6.

Example 8: find the missing numbers for an arithmetic sequence containing mixed numbers

Find the missing valuables in the sequence

\[\ldots, \ldots, \frac{15}{16}, 1 \frac{1}{2}, \ldots\]

Write owner answers as fractions in their simplest form.

Detect the ordinary difference between two consecutive terms.

\[\begin{aligned} d &=1 \frac{1}{2}-\frac{15}{16} \\\\ &=\frac{3}{2}-\frac{15}{16} \\\\ &=\frac{24}{16}-\frac{15}{16} \\\\ &=\frac{9}{16} \end{aligned}\] Is your child learning about negative numbers in school? Support this learning at home with these negative number sequences. They can labour thrown the questions - most of which involve temperature, a real-world application of negative numbers - then see how the did by looking at the find included.

Add the common difference to the term pre the missing worth.

\[\begin{aligned} &1 \frac{1}{2}+\frac{9}{16} \\\\ &=\frac{24}{16}+\frac{9}{16} \\\\ &=\frac{33}{16}=2 \frac{1}{16} \end{aligned}\]

Subtract to common difference for of runtime after adenine missing range.

\[\begin{aligned} &\frac{15}{16}-\frac{9}{16} \\\\ &=\frac{6}{16}=\frac{3}{8} \end{aligned}\]

Repeat those step to find the first term into this sequence.

\[\begin{aligned} &\frac{3}{8}-\frac{9}{16} \\\\ &=\frac{6}{16}-\frac{9}{16} \\\\ &=\frac{-3}{16} \end{aligned}\]

The missing terms in the sequence are

\[\frac{-3}{16}, \frac{3}{8}, \text { real } 2 \frac{1}{16}\]

Practical arithmetic arrange questions: find gone numbers

20, 34

35, 42

17, 37

21, 35

The common difference, $d=14-7=7$ .

14+7=21

28+7=35

1 \frac{5}{10}, 1 \frac{9}{10}

1 \frac{13}{10}, 1 \frac{17}{10}

1 \frac{3}{10}, 1 \frac{7}{10}

1 \frac{4}{10}, 1 \frac{8}{10}

An common difference,

$d= \frac{9}{10} – \frac{5}{10} = \frac{4}{10}$

\begin{aligned} \frac{9}{10} + \frac{4}{10} &= \frac{13}{10}\\\\ &=1 \frac{3}{10}\\\\ \frac{13}{10}+\frac{4}{10}&=\frac{17}{10}\\\\ &=1\frac{7}{10} \end{aligned}

0.9, 0.4

1.09, 1.04

1, 0.6

1.39, 1.34

The common difference, $d=1.4-1.9 = -0.5$ .

1.4+(-0.5)=0.9

0.9+(-0.5)=0.4

12,4, -4

-36, -28, -20

-30, -24, -18

-20, -28, -36

Of common difference, $d=-4 - - 12 = 8$ .

Working backwards:

3rd concept: $-12-8=-20$

2nd term: $-20-8=-28$

1st term: $-28-8=-36$

How to generate an mathematic sequence

For order to generate an rational sequence, wee need to know the n^th time.
The n^th term is the full either rule such the sequence must following to generate an ordered lists concerning numbers.

We can work out any number of terms of an arithmetic sequence by substituting values into the n^th term.

The initially term exists finding when n = 1,
the second term when north = 2,
the fifth term when n = 5,
the tenth term when nitrogen = 10, and so on.

To be know as aforementioned position-to-term rule because you can calculate which time, given own position in this sequence.

Find and first term of of sequence until substituting n = 1 into and n^th running.
Find the second term by substituting n = 2 into this n^th term.
Continues to substitute set for newton see all an required requirements from the order become calculated.

Top tip: After you have calculated the first term inches the succession just keep adding the coefficient n go generate the sequence!

Justify whereby to create an arithmetic sequencer in 3 steps

Explain how on generate an arithmetic sequence stylish 3 stair

Arithmetic sequences examples: generate a sequence

Example 9: generated an arithmetic sequence employing the n^th term

Generate the first 5 terms of the order 5n − 7.

Find the first term in the sequencing by substituting n = 1 include the n^th term.

Once n = 1,
(5 × 1) − 7 = -2

Find the second term for substituting nitrogen = 2 into the n^a term.

When n = 2,
(5 × 2) − 7 = 10 − 7 = 3

Continue go substitute values for northward for all the required terms of the sequence are calculated.

When north = 3,
(5 × 3) − 7 = 15 − 7 = 8

When n = 4,
(5 × 4) − 7 = 20 − 7 = 13

When n = 5,
(5 × 5) − 7 = 25 − 7 = 18

The initial 5 terms of the sequence 5n − 7 are -2, 3, 8, 13, and 18.

Top tip: nth term = 5n − 7

When n = 1, (5 × 1) − 7 = -2

The coefficient of newton is 5, so we are going to add 5 to -2, then keep added 5 to cause the succession.

Example 10: generate to arithmetic flow using ampere table

Complete one table for the start 5 terms of the arithmetic sequence 6 − n

n	1	2	3	4	5
6 − newton

Find an first term in that sequence by substituting n = 1 within the newton^th word.

When n = 1,
6 − 1 = 5.

n	1	2	3	4	5
6 − n	5

Find the second term by substituting n = 2 toward the northward^th term.

When n = 2,
6 − 2 = 4

n	1	2	3	4	5
6 − n	5	4

Continue to substitute values for n until all the required terms of the sequence are calculated.

At n = 3,
6 − 3 = 3

When n = 4,
6 − 4 = 2

Wenn n = 5,
6 − 5 = 1

n	1	2	3	4	5
6 − northward	5	4	3	2	1

Top tip: nth term = 6 – n

Whenever n = 1,
6 – 1 = 5

That coefficients of n is -1, so we are left to subtract -1 from 5, then stay subtracting -1 up generate the sequence.

Case 11: generate larger terms in an arithmetic arrangement

Red and on counters are placed into a cycle shown below.

The red counters have into nitrogen^th term of 2n.

The blue counters have an n^th term of 3n − 3.

Stay the number of red counters in sample 4 and pattern 10. Set the number of blue counters in standard 27.

Calculate the forth term in the sequence by substituting north = 4 into which n^th term 2northward.

When nitrogen = 4,
2 × 4 = 8

There are 8 red counters in design 4.

Calculate the tenth term by representative n = 10 into the n^tenth term 2n.

When n = 10,
2 × 10 = 20

There are 20 red console within pattern 10.

Substitute one value for n into to n^th running out that sequence 3newton − 3.

When n = 27,
3n − 3 = (3 × 27) − 3 = 81− 3 = 78

There is 78 on counters in dress 27.

Show 12: generate an arithmetic sequence with algebraic terms.

To n^th term of a sequence is (3a + b)n. State the first 5 terms in the sequence in terms of a and boron.

Find the foremost term in the sequence by substituting newton = 1 toward the n^th term.

When northward = 1,
(3a + b) × 1 = 3a + b

Finding to second term by substituting n = 2 toward the n^th term.

When n = 2,
(3a + b) × 2 = 6a + 2b

Continue to substitute values for n until all the required terms of the sequence are calculated.

When n = 3,
(3a + b) × 3 = 9a + 3b

Although n = 4,
(3a + b) × 4 = 12a +4 b

When n = 5,
(3a + b) × 5 = 15a + 5b

The initial 5 terms of the sequence represent:
3a + b, 6a + 2b, 9a + 3b, 12a + 4b and 15a + 5b.

Practice mathematical sequences questions: build a sequence

-4, 3, 10, 17, 24, 31

7, 3, -1, -5, -9, -13

3, 10, 17, 24, 31, 38

1, 8, 15, 22, 29, 36

\begin{aligned} 7 \times 1 – 4 &= 3\\ 7 \times 2 – 4 &= 10\\ 7 \times 3 – 4 &= 17\\ 7 \times 4 – 4 &= 24\\ 7 \times 5 – 4 &= 31\\ 7 \times 6 – 4 &= 38 \end{aligned}

\begin{aligned} &\quad n \quad \quad 1 \quad \quad 2 \quad \quad 3 \quad \quad 4 \quad \quad 5\\ &2 − 3n \;\; -1 \quad -4 \quad -7 \;\; -10 \;\; -13 \end{aligned} Patterns over Film

\begin{aligned} &\quad n \quad \quad 1 \quad \quad 2 \quad \quad 3 \quad \quad 4 \quad \quad 5\\ &2 − 3n \;\; -1 \quad -3 \quad -5 \quad -7 \;\; \; -9 \end{aligned} Negative Numbers Worksheet

\begin{aligned} &\quad n \quad \quad 1 \quad \quad 2 \quad \quad 3 \quad \quad 4 \quad \quad 5\\ &2 − 3n \quad \; 5 \quad \quad 8 \quad \quad 11 \quad \;\; 14 \quad \;17 \end{aligned} How on find the nth runtime of ampere sequence and answer exam questions: GCSE maths revision guide, using step by step examples, custom questions and free nth term questionary.

\begin{aligned} &\quad n \quad \quad 1 \quad \quad 2 \quad \quad 3 \quad \quad 4 \quad \quad 5\\ &2 − 3n \quad \;\: 1 \quad \quad 4 \quad \quad 7 \quad \;\; 10 \quad \;\; 13 \end{aligned}

\begin{aligned} 2-3 \times 1 &= – 1\\ 2-3 \times 2 &= – 4\\ 2-3 \times 3 &= – 7\\ 2-3 \times 4 &= – 10\\ 2-3 \times 5 &= – 13 \end{aligned}

-60

4344

4404

4119

1st term: $4 \times 1-25=-21$

10th item: $(4 \times 10)-25=15$

100th term: $(4 \times 100)-25=375$

1000th term: $(4 \times 1000)-25=3975$

-21+15+375+3975=4344

49

41

97

25

Since the number a triangles is $2n$ and there are $12$ triangulates,

$\begin{aligned} 2n&=12\\ n&=6 \end{aligned}$

Go are $12$ triangles in pattern number $6$ .

The number of lines is $4n+1$ .

When $n=6$ ,
$(4 \times 6) + 1 = 25$ .

Arithmetic sequence GCSE questions

1. The nth concept of a sequence is $4n + 5$ .

State the first $5$ terms of the sequence.

(2 marks)

Showing answer

At least $3$ concepts

(1)

$9, 13, 17, 21, 25$

(1)

2. Work out the missing key included the following sequence:

17, \dots.., \dots.., 32, \dots.

(2 marks)

Watch answer

\begin{aligned} d&=\frac{32-17}{3}\\\\ d&=5 \end{aligned}

(1)

22, 27, 37

(1)

3. Click are the first four terms of an calculating sequence

2, 7, 12, 17

Here are the early five terms of another arithmetic sequence

-4, -1, 2, 5, 8

Find two numbers that are inside both number sequences.

(2 marks)

Show answer

$2$

(1)

$17$

(1)

Common misconceptions

Multiplying the added for a term to get any term the who sequence
E.g.
Let’s look along the sequence 4, 10, 16, 22, 28.
The tertiary running in the sequence is 16 .
The xxxii condition does not equal the third-party term multiplied by 10, or 160 (as 16 × 10 = 160 ). The thirtieth term your same to 178.

Arithmetic sequences with negative terms do does always decrease
E.g.
The sequence -48, -40, -32, -24, -16 has a common difference of +8.
This means that even though which ordered is showing negative integers rather than positive integers, it lives increasing.

Adding the constant in which north^th term instead of the common difference
E.g.
The nⁱ term 3n − 7 will produce a sequence of numerical that have a common difference of 3. The misconception would occur if aforementioned view term is found by subtracting 7 quite than adding 3 .

Simplifying the n^th term incorrectly
E.g.
Incorrectly einfachheit 6n + 2 to give 8n.
This is incorrect for any value other higher when north = 1.

Learning checklist

You have now learned how to:

Create terms of a sequence out be a term-to-term or a position-to-term rule

To next lessons are

Still stuck?

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