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S1–S4 Mathematics A3 Sequences

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**A3.1 Special number patterns**

Contents A3 Sequences A A3.1 Special number patterns A A3.2 Describing and continuing sequences A A3.3 Generating sequences from rules A A3.4 Finding the nth term of a linear sequence A A3.5 Sequences from practical contexts

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**Even numbers Even numbers are numbers that are exactly divisible by 2.**

For example, 48 is an even number. It can be written as 48 = 2 × 24. All even numbers end in 0, 2, 4, 6 or 8. Even numbers can be illustrated using dots or counters arranged as follows: Numbers that can be illustrated using patterns of evenly spaced dots or counters are sometimes called figurate numbers. Pupils no not need to remember this term, however. Examples of figurate numbers include even numbers, odd numbers, triangular numbers, square numbers and cube numbers. Illustrating numbers using patterns of dots help us to deduce and justify their properties. Explain that even numbers can also be negative, even though these negative numbers cannot be shown as patterns of dots. 2 4 6 8 10

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**Odd numbers Odd numbers leave a remainder of 1 when divided by 2.**

For example, 17 is an odd number. It can be written as 17 = 2 × 8 + 1 All odd numbers end in 1, 3, 5, 7 or 9. Odd numbers can be illustrated using dots or counters arranged as follows: Explain that odd numbers can also be negative even though they cannot be shown figuratively in this way. 1 3 5 7 9

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Making triangles Ask pupils how to arrange counters into triangles by putting consecutive numbers of counters into rows. For example, we can put one counter at the top, two counters in the next row and three counters in the third. Demonstrate this arrangement using round counters on the board. There are six counters in this arrangement. Six is a triangular number. What is the next triangular number? Show that we can find the next triangular number by adding a row of four counters. The next triangular number is ten.

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Triangular numbers Triangular numbers are numbers that can be written as the sum of consecutive whole numbers starting with 1. For example, 15 is a triangular number. It can be written as 15 = Triangular numbers can be illustrated using dots or counters arranged in triangles: 15 10 6 3 1

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Making squares Ask volunteers to come to the board and arrange 9 counters to make a square. Tell pupils that the number 9 is a square number. Ask, Can we arrange 16 tiles to make a square? Ask a volunteer to demonstrate this, and ask pupils to tell you how we can decide whether or not 16 is a square number. Establish that to make a square we need the same number of rows and columns. We can arrange 16 counters in 4 rows and 4 columns and so it is a square number. It follows that we can make a square number by multiplying a whole number by itself. Ask, Can we arrange 20 counters in rows and columns to make a square? What is the next square number? (25) What is the smallest square number? (1) Tell pupils that when we multiply a number by itself, for example 6 × 6, it is called squaring the number and that another way of saying 6 × 6 is ‘six squared’. The result, for example, 36, is called a square number.

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Square numbers Square numbers are obtained when a whole number is multiplied by itself. They are sometimes called perfect squares. For example, 49 is a square number. It can be written as 49 = 7 × 7 or 49 = 72. Square numbers can be illustrated using dots or counters arranged in squares: 25 Pupils should be able to recognise square numbers up to 122 = 144. 16 9 4 1

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Making square numbers There are several ways to generate a sequence of square numbers. We can multiply a whole number by itself. We can add consecutive odd numbers starting from 1. We can add together two consecutive triangular numbers.

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**Adding consecutive odd numbers**

The seventh square number is 49. The eighth square number is 64. The ninth square number is 81. The sixth square number is 36. The tenth square number is 100. The fifth square number is 25. The second square number is 4. The first square number is 1. The fourth square number is 16. The third square number is 9. By using a different colour for the counters that are added on between consecutive square numbers we can see how square numbers can be made by adding consecutive odd numbers starting from 1. Pupils could investigate this result before seeing it on this slide. = 64 = 81 = 100 = 49 = 16 1 + 3 = 4 = 9 = 25 = 36

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**Adding two consecutive triangular numbers**

We can also make square numbers by adding two consecutive triangular numbers. Again, pupils could investigate this result before seeing it on this slide. = 81 = 100 = 64 = 36 1 + 3 = 4 3 + 6 = 9 = 16 = 25 = 49

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Powers of 2 Powers of 2 are found by multiplying the number 2 by itself a given number of times. For example, 2 × 2 × 2 = 23 = 8. We can show powers of two like this: 25 = 32 26 = 64 23 = 8 24 = 16 We can show this sequence of powers of 2 like this. Notice that the numbers get bigger very quickly. Ask: What is the difference between the arrangements for odd powers, 21, 23, and 25, and the arrangements for even powers 22, 24, 26? Establish that the odd powers are arranged in columns containing two squares and the even powers are arranged in squares. Deduce from this that every even power of 2 is a square number and every odd power of two is double a square number. 21 = 2 22 = 4 Each number is double the one before it.

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Cube numbers Cube numbers are obtained when a whole number is multiplied by itself and then by itself again. For example, 64 is a cube number. It can be written as 64 = 4 × 4 × 4 or 64 = 43. Cube numbers can be illustrated using spheres arranged in cubes: Count the number of spheres in each arrangement by multiplying the number of cubes in each layer by the total number of layers. Link to volume = length × width × height. 1 8 27 64 125

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Multiples A multiple of a number is found by multiplying the number by a whole number. For example, to find the first five multiples of 3 multiply 3 by 1, 2, 3, 4, and 5 in turn. Multiples of 3 can be illustrated using dots or counters arranged in rectangles: Ask pupils if the multiples of all numbers can be arranged in rectangles. 3 6 9 12 15

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Primes A prime number is defined as a whole number that has two, and only two, factors. These two factors are the number itself and 1. The first 10 prime numbers are: 2 3 5 7 11 13 17 19 23 29

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Name this sequence Use this activity to revise the names of some known (and perhaps unknown) sequences. For example, tetrahedral numbers are formed by summing consecutive triangular numbers. The can be illustrated by arranging spheres in tetrahedral arrangements. The first five tetrahedral numbers are 1, 4, 10, 20 and 35. For each sequence pupils could also be asked to give the next few terms.

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**Squares, triangles and primes**

Discuss how the highest possible score can be achieved.

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**A3.2 Describing and continuing sequences**

Contents A3 Sequences A A3.1 Introducing sequences A A3.2 Describing and continuing sequences A A3.3 Generating sequences from rules A A3.4 Finding the nth term of a linear sequence A A3.5 Sequences from practical contexts

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**Introducing sequences**

In maths, we call a list of numbers in order a sequence. Each number in a sequence is called a term. 4, 8, 12, 16, 20, 24, 28, 32, ... 1st term 6th term If terms are next to each other they are referred to as consecutive terms. When we write out sequences, consecutive terms are usually separated by commas.

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**Sequences that increase in equal steps**

We can describe sequences by finding a rule that tells us how the sequence continues. To work out a rule it is often helpful to find the difference between consecutive terms. For example, look at the difference between each term in this sequence: 3, , , , , , , , ... +4 +4 +4 +4 +4 +4 +4 Remind pupils that the sequence of multiples of 4 increases by adding 4 each time. Ask pupils how this sequence is related to multiples of 4. This will help pupils find the nth term of this type of sequence in latter work. This sequence starts with 3 and increases by 4 each time. Every term in this sequence is one less than a multiple of 4.

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**Sequences that decrease in equal steps**

Can you work out the next three terms in this sequence? 22, 16, , , –2, –8, –14, –20, ... –6 –6 –6 –6 –6 –6 –6 How did you work these out? This sequence starts with 22 and decreases by 6 each time. This example shows that sequences can also continue by subtracting the same amount each time. Remind pupils that they must check that every number in the sequence obeys the same rule. Sequences that increase or decrease in equal steps are called linear or arithmetic sequences.

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**Sequences that increase in increasing steps**

Some sequences increase or decrease in unequal steps. For example, look at the differences between terms in this sequence: 5, , , , , , , , ... +1 +2 +3 +4 +5 +6 +7 Again, remind pupils that they must check that every number in the sequence obeys the same rule. This sequence starts with 5 and increases by 1, 2, 3, 4, … The differences between the terms form a linear sequence.

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**Sequences that decrease in decreasing steps**

Can you work out the next three terms in this sequence? 7, , , , 6, 5.5, 4.9, 4.2, ... –0.1 –0.2 –0.3 –0.4 –0.5 –0.6 –0.7 How did you work these out? This sequence starts with 7 and decreases by 0.1, 0.2, 0.3, 0.4, 0.5, … Explain that a second row of differences shows the differences between the differences. In this example, the difference between each difference is –0.1. With sequences of this type it is often helpful to find a second row of differences.

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**Using a second row of differences**

Can you work out the next three terms in this sequence? 1, , , , , 41, 58, 78, ... +2 +5 +8 +11 +14 +17 +20 +3 +3 +3 +3 +3 +3 Look at the differences between terms. A sequence is formed by the differences so we look at the second row of differences. This shows that the differences increase by 3 each time.

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**Sequences that increase by multiplying**

Some sequences increase or decrease by multiplying or dividing each term by a constant factor. For example, look at this sequence: 2, , , , , , , 256, ... ×2 ×2 ×2 ×2 ×2 ×2 ×2 This sequence starts with 2 and increases by multiplying the previous term by 2. This sequence increases in unequal steps. It is not linear. If we looked at the differences between the consecutive terms they would be 2, 4, 8, 16, 32 … in other words, they would form the same sequence. This is a sequence of powers of 2. We could write it as 21, 22, 23, 24, 25, 26, 27, … All of the terms in this sequence are powers of 2.

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**Sequences that decrease by dividing**

Can you work out the next three terms in this sequence? 1 4 , , 1 16 1024, 256, 64, 16, , 1, ÷4 How did you work these out? This sequence starts with 1024 and decreases by dividing by 4 each time. Each term in this sequence is one quarter of the term before. We could also continue this sequence by multiplying by each time, since that is equivalent to dividing by 4. 1 4

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**Fibonacci and Fibonacci-type sequences**

Can you work out the next three terms in this sequence? 1, 1, 2, 3, 5, , , 21, 34, 55, ... 1 + 1 1 + 2 2 + 3 3 + 5 5 + 8 8 + 13 How did you work these out? This sequence starts with 1, 1 and each term is found by adding together the two previous terms. Sequences of this type must be generated by two numbers. Ask pupils if this type of sequence can be descending, for example if the sequence is generated by two negative numbers. As an extension activity, pupils could investigate this sequence for a variety of different starting numbers. The Fibonacci sequence appears in many situations in nature. Ask pupils to research some examples on the Internet. This sequence is called the Fibonacci sequence after the Italian mathematician who first wrote about it.

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**Describing and continuing sequences**

Here are some of the types of sequence you may come across: Sequences that increase or decrease in equal steps. These are called linear or arithmetic sequences. Sequences that increase or decrease in unequal steps by multiplying or dividing by a constant factor. Sequences that increase or decrease in unequal steps by adding or subtracting increasing or decreasing numbers. Discuss each of these different types of sequences asking pupils to give examples for each one. The second type of sequence is a sequence of powers or geometric sequence, the third type is a quadratic sequence, and the fourth is a Fibonacci-type sequence. These can be broadly divided into two types: Sequences that increase or decrease in equal steps – linear sequences – and sequences that increase or decrease in unequal steps – non-linear sequences. Ask pupils how we can use the differences between consecutive terms to help us to recognize each type. In the first type of sequence the differences are constant. In the second type of sequence the differences form another geometric sequence. In the third type the differences form a linear sequence and so the second row of differences are constant. In the fourth type the differences form the same sequence. Sequences that increase or decrease by adding together the two previous terms.

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**Continuing sequences A number sequence starts as follows 1, 2, ...**

How many ways can you think of continuing the sequence? Examples are: 1, 2, 3, 4, 5, … Adding on 1 each time, 1, 2, 4, 8, 16, … Doubling the number each time, 1, 2, 4, 7, 11, … Add on 1, then 2, then 3, … 1, 2, 3, 5, 8, … Add together the previous two terms etc. Ensure that pupils are starting to use the correct mathematical language to describe their rules. Give the next three terms and the rule for each one.

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Finding missing terms Discuss strategies for finding the missing terms in each linear sequence.

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Sequence grid Start the activity by asking pupils to work out the number that is hidden by the orange counter. Initially, do not tell pupils how the numbers in the grid have been generated. Ask pupils if they can see any patterns. The numbers in the rows, columns and diagonals form arithmetic sequences. Pupils will need to determine the rule for two of these sequences by subtracting numbers in adjacent squares (either horizontally, vertically or diagonally) and counting on. You may click on any counter at any time to reveal the number beneath it. As an extension or homework activity ask pupils to fill in their own sequence grids. They can generate the table by choosing one rule for the rows and another for the columns. They can then investigate how the sequences in the diagonals are related to the sequences in the rows and columns.

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**A3.3 Generating sequences from rules**

Contents A3 Sequences A A3.1 Introducing sequences A A3.2 Describing and continuing sequences A A3.3 Generating sequences from rules A A3.4 Finding the nth term of a linear sequence A A3.5 Sequences from practical contexts

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**Generating sequences from flow charts**

A sequence can be given by a flow chart. For example, START Write down 3. This flow chart generates the sequence: 3, 4.5, 6, 6.5, 9, 10.5. Add on 1.5. Write down the answer. This sequence has only six terms. Click to reveal each section of the flow chart. Follow the instructions until you get to 10.5 and then click to reveal the yes arrow and the instruction to stop. Is the answer more than 10? No It is finite. Yes STOP

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**Generating sequences from flow charts**

START Write down 5. This flow chart generates the sequence 5, 2.9, 0.8, –1.3, –3.4. Subtract 2.1. Write down the answer. Is the answer less than –3? No Go through this flow chart in the same way. Yes STOP

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**Generating sequences from flow charts**

START Write down 3 and 4. This flow chart generates the sequence 3, 4, 7, 11, 18, 29, 47, 76, 123. Add together the two previous numbers. Write down the answer. Is the answer more than 100? No Yes STOP

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**Predicting terms in a sequence**

Usually, we can predict how a sequence will continue by looking for patterns. For example, 87, 84, 81, 78, ... We can predict that this sequence continues by subtracting 3 each time. However, sequences do not always continue as we would expect. Discuss ways in which the sequence 1, 2, 4 might continue. For example, by doubling the previous term or by adding 1, adding 2, adding 3 etc.. For example, A sequence starts with the numbers 1, 2, 4, ... How could this sequence continue?

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Continuing sequences Here are some different ways in which the sequence might continue: 1 2 4 7 11 16 22 +1 +2 +3 +4 +5 +6 1 2 4 8 16 32 64 ×2 ×2 ×2 ×2 ×2 ×2 We can never be certain how a sequence will continue unless we are given a rule or we can justify a rule from a practical context.

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**Continuing sequences This sequence continues by adding 3 each time. 1**

4 7 10 13 16 19 +3 +3 +3 +3 +3 +3 We can say that rule for getting from one term to the next term is add 3. This is called the term-to-term rule. The term-to-term rule for this sequence is +3.

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**Using a term-to-term rule**

Does the rule +3 always produce the same sequence? No, it depends on the starting number. For example, if we start with 2 and add on 3 each time we have 2, 5, 8, 11, 14, 17, 20, 23, ... There are infinitely many sequences that follow the rule ‘add three’. Encourage pupils to think of different possible starting numbers. The starting number could be a fraction, a negative number or a very large number. Ask pupils if it is possible to find a sequence that continues by adding 3 where: a) all the numbers are multiples of 3? b) all the numbers are odd? c) all the numbers are multiples of 9? If we start with 0.4 and add on 3 each time we have 0.4, 3.4, 6.4, 9.4, 12.4, 15.4, 18.4, 21.4, ...

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**Writing sequences from term-to-term-rules**

A term-to-term rule gives a rule for finding each term of a sequence from the previous term or terms. To generate a sequence from a term-to-term rule we must also be given the first number in the sequence. For example, 1st term Term-to-term rule 5 Add consecutive even numbers starting with 2. This gives us the following sequence: 5 7 11 17 25 35 47 ... +2 +4 +6 +8 +10 +12

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**Sequences from term-to-term rules**

Write the first five terms of each sequence given the first term and the term-to-term rule. 1st term Term-to-term rule First five terms –7 Add 3 –7, –4, –1, 2, 5 100 Subtract 8 100, 92, 84, 76, 68 3 Double 3, 6, 12, 24, 48 0.04 Edit the numbers in this slide to produce more or less challenging examples. Multiply by 10 0.04, 0.4, 4, 40, 400 1.3 Subtract 0.6 1.3, 0.7, 0.1, –0.5, –1.1 111 Divide by 5 111, 22.2, 4.44, 0.888, 0.1776

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**Sequences from position-to-term rules**

Sometimes sequences are arranged in a table like this: Term nth … 6th 5th 4th 3rd 2nd 1st Position 3 6 9 12 15 18 … 3n We can say that each term can be found by multiplying the position of the term by 3. This is called a position-to-term rule. Stress that once we know a position-to-term rule we can find any term in the sequence given its position in the sequence. Ask pupils to give other terms in the sequence with the position-to-term rule 3n. For example, what is the 15th term in the sequence? You could also ask pupils to give you the position of a given term in the sequence using inverse operations. For example, 42 is a term in this sequence. What position is it in? For this sequence we can say that the nth term is 3n, where n is a term’s position in the sequence. What is the 100th term in this sequence? 3 × 100 = 300

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**Sequences from position-to-term rules**

Start by revealing the value of the nth term and ask pupils to find the value of each term using substitution. You may choose from linear and quadratic sequences. Alternatively, give pupils the six terms and ask then to find an expression for the nth term.

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**Sequence generator – linear sequences**

Use this activity to review finding position-to-term rules and term-to-term rules for simple arithmetic sequences. Click on the position-to-term rule to edit it.

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**Sequence generator – non-linear sequences**

Use this activity to generate and investigate non-linear sequences. The pen tool can be used to find patterns in the differences between consecutive terms. Reveal then click on the position-to-term rule to edit it.

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**The position-to-term rule?**

Sequences and rules Which rule is best? The term-to-term rule? The position-to-term rule? Discuss the difference between the term-to-term rule and the position-to-term rule for a sequence. Some may say that the term-to-term rule is better because it doesn’t use algebra. Conclude that the position-to-term rule is usually more useful because usually allows you to find the value of any term in the sequence given its position. For example, if you want to know the 100th term in a sequence you don’t have to calculate the first 99 terms to find it. The position-to-term rule is usually written algebraically. We call it the rule for the ‘nth term’. Explain that the ‘nth term’ of a sequence can also be called the ‘general term’ of the sequence. The general term gives us a rule which is true for every number in the sequence using algebra. For example, the general term of the sequence 5, 10, 15, 20, 25, ... is 5n (where 5 is the first term). Point out that for some sequences that increase in unequal steps, the position-to-term rule can be very complicated to find and use, for example, in The Fibonacci Sequence where each term is the sum of the previous two terms. For such sequences the term-to-term rule is best.

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**A3.4 Finding the nth term of a linear sequence**

Contents A3 Sequences A A3.1 Introducing sequences A A3.2 Describing and continuing sequences A A3.3 Generating sequences from rules A A3.4 Finding the nth term of a linear sequence A A3.5 Sequences from practical contexts

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**Sequences of multiples**

All sequences of multiples can be generated by adding the same amount each time. They are linear sequences. For example, the sequence of multiples of 5: 5, , , , , , 35, , … +5 +5 +5 +5 +5 +5 +5 can be found by adding 5 each time. Compare the terms in the sequence of multiples of 5 to their position in the sequence: Reveal the sequence of multiples of 5 and ask pupils if they can you find a rule linking the position of each term to the value of the term. Position 1 5 2 10 3 15 4 20 5 25 … n × 5 × 5 × 5 × 5 × 5 × 5 Term 5n

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**Sequences of multiples**

The sequence of multiples of 3: 3, , , , , , , , … +3 +3 +3 +3 +3 +3 +3 can be found by adding 3 each time. Compare the terms in the sequence of multiples of 3 to their position in the sequence: Stress that for any sequence of multiples the nth term will be the difference between the consecutive terms (or d ) multiplied by n. Position 1 3 2 6 3 9 4 12 5 15 … n ×3 ×3 ×3 ×3 ×3 ×3 Term 3n

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**Sequences of multiples**

The nth term of a sequence of multiples is always dn, where d is the difference between consecutive terms. For example, The nth term of 4, 8, 12, 16, 20, 24, … is 4n The 10th term of this sequence is 4 × 10 = 40 We can work out terms in the sequence by multiplying the position number by 4. The 25th term of this sequence is 4 × 25 = 100 The 47th term of this sequence is 4 × 47 = 188

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**Finding the nth term of a linear sequence**

The terms in this sequence 4, , , , , , , , … +3 +3 +3 +3 +3 +3 +3 can be found by adding 3 each time. Compare the terms in the sequence to the multiples of 3. Position 1 2 3 4 5 … n × 3 × 3 × 3 × 3 × 3 × 3 Multiples of 3 Explain that this sequence increases by 3 each time like the 3 times table (multiples of 3, in other words). We therefore compare the position numbers to multiples of 3. 3 6 9 12 15 3n + 1 + 1 + 1 + 1 + 1 + 1 Term 4 7 10 13 16 … 3n + 1 Each term is one more than a multiple of 3.

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**Finding the nth term of a linear sequence**

The terms in this sequence 1, , , , , , , , … +5 +5 +5 +5 +5 +5 +5 can be found by adding 5 each time. Compare the terms in the sequence to the multiples of 5. Position 1 2 3 4 5 … n × 5 × 5 × 5 × 5 × 5 × 5 Multiples of 5 Explain that this sequence increases by 5 each time like the 5 times table (multiples of 5, in other words). We therefore compare the position numbers to multiples of 5. 5 10 15 20 25 5n – 4 – 4 – 4 – 4 – 4 – 4 Term 1 6 11 16 21 … 5n – 4 Each term is four less than a multiple of 5.

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**Finding the nth term of a linear sequence**

The terms in this sequence 5, , , –1, –3, –5, –7, –9, … –2 –2 –2 –2 –2 –2 –2 can be found by subtracting 2 each time. Compare the terms in the sequence to the multiples of –2. Position 1 2 3 4 5 … n × –2 × –2 × –2 × –2 × –2 × –2 Multiples of –2 Explain that this sequence decreases by 2 each time. We therefore compare the position numbers to multiples of –2. Explain that it is preferable not to have a negative sign at the beginning of an expression. We therefore write –2n + 7 as 7 – 2n. –2 –4 –6 –8 –10 –2n + 7 + 7 + 7 + 7 + 7 + 7 Term 5 3 1 –1 –3 … 7 – 2n Each term is seven more than a multiple of –2.

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**Finding the nth term of a linear sequence**

Use this activity to review finding the nth tern of a linear sequences. Click on the blue arrows to generate a random sequence. A pupil could also be asked to do this. Reveal any three consecutive terms and ask pupils to find the remaining terms. By finding the common difference, or otherwise, ask pupils to find the rule for the nth term. Make the activity more difficult by revealing three non-consecutive terms. Establish that the nth term of any linear sequence can be written as dn + (a – d), where d is the difference between two consecutive terms and a is the value of the first term. (a – d) is the value of the 0th term. The value of the 0th term has been included so that link between its value and the rule for the nth term can be established. Once the rule has been revealed check by substitution that all of the terms fit the rule.

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**A3.5 Sequences from practical contexts**

Contents A3 Sequences A A3.1 Introducing sequences A A3.2 Describing and continuing sequences A A3.3 Generating sequences from rules A A3.4 Finding the nth term of a linear sequence A A3.5 Sequences from practical contexts

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**Sequences from practical contexts**

The following sequence of patterns is made from L-shaped tiles: Number of Tiles 4 8 12 16 Ask pupils to tell you what type of sequence is generated by this tiling pattern. Establish that it is an linear sequence because the same number of tiles is added on each time. (It is also the sequence of multiples of 4). Ask pupils, How do you know that 4 tiles will be added on to make the next pattern? Tell pupils that we need to explain why four tiles are added on each time by looking at how the pattern is built up. This is called a justification of the rule. The number of tiles in each pattern forms a sequence. How many tiles will be needed for the next pattern? We add on four tiles each time. This is a term-to-term rule.

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**Sequences from practical contexts**

A possible justification of this rule is that each shape has four ‘arms’ each increasing by one tile in the next arrangement. The pattern give us multiples of 4: 1 lot of 4 Explain that 4n is the position-to-term rule or rule for the nth term. It can also be called the general term of the sequence. 2 lots of 4 3 lots of 4 4 lots of 4 The nth term is 4 × n or 4n. Justification: This follows because the 10th term would be 10 lots of 4.

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**Sequences from practical contexts**

Now, look at this pattern of blocks: Number of Blocks 4 7 10 13 How many blocks will there be in the next shape? We add on 3 blocks each time. Again stress that rules generated from practical situations, such as patters made from blocks, must be justified by referring to the context from which the pattern is generated. This is the term-to term rule. Justification: The shapes have three ‘arms’ each increasing by one block each time.

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**Sequences from practical contexts**

How many blocks will there be in the 100th arrangement? We need a rule for the nth term. Look at pattern again: 1st pattern 2nd pattern 3rd pattern 4th pattern Remember, the rule for the nth term will allow us to work out the value of any term in the sequence by simply substituting the given term’s position number into the rule. Look at the patterns and establish verbally that to get the number of blocks we must times the pattern’s position number by 3 and then add one for the middle block. The nth pattern has 3n + 1 blocks in it. Justification: The patterns have 3 ‘arms’ each increasing by one block each time. So the nth pattern has 3n blocks in the arms, plus one more in the centre.

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**Sequences from practical contexts**

So, how many blocks will there be in the 100th pattern? Number of blocks in the nth pattern = 3n + 1 When n is 100, Number of blocks = (3 × 100) + 1 = 301 How many blocks will there be in the: a) 10th pattern? (3 × 10) + 1 = 31 Use substitution to find terms given their pattern number. A pattern contains 37 blocks. What pattern number is it? Explain that we can work this out using inverse operations. To get the number of blocks given the pattern number we multiply by 3 and add 1. Using inverse operations in reverse order, to get the pattern number given the number of blocks we subtract 1 and divide by 3. (37 – 1)/3 is 12, so the 12th pattern contains 37 blocks. b) 25th pattern? (3 × 25) + 1 = 76 c) 52nd pattern? (3 × 52) + 1 = 157

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Paving slab patterns Explain that a gardener is experimenting with different patterns for a path. Start by investigating pattern 1 by dragging the arrow and revealing the number of blue and yellow tiles. Ask pupils if they can spot any patterns. Establish that for this pattern there are 8 blue tiles for every yellow tile. We can therefore find the number of blue tiles by multiplying the number of yellow tiles by 8. Investigate the remaining patterns in the same way. For each pattern, pupils could put the number of yellow tiles and the corresponding number of blue tiles into a table. By using y for the number of yellow tiles, they could then find a rule giving the number of blue tiles in terms of the number of yellow tiles. Alternatively, they could write the number of blue tiles in terms of n, the pattern number.

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Dotty patterns Explain that for each shape in the pattern we can write down the number of dots that we pass through and the number of lines connecting the dots. These results can then be put into a table. Use the slider to demonstrate different numbers of shapes and ask pupils to tell you the number of dots and the number of lines used. Enter these values into the table by clicking on empty cells. After a few values have been entered ask pupils to predict the remaining values. Ask pupils how they made their predictions. For pattern 1, establish that for every shape we add on, we add another 3 dots and another 4 lines. Encourage pupils to justify these rules within the context of the pattern. The term-to-term rule for the sequence formed by the number of dots is ‘add 3’ and the term-to-term rule for the sequence formed by the number of lines is ‘add 4’. Ask pupils, How many dots will there be if n shapes are drawn? Remind pupils again that that n can be any whole number. Ask them, What do you do to the number of shapes, n, to get the number of dots? Establish that since the sequence is formed by adding 3 each time, like the three times table, the rule for the nth term must be based on 3n. Ask pupils to mentally multiply some of the shape numbers by 3 and ask them what they need to add on to get the number of dots. For example, if there are 5 shapes, three times 5 is 15. We then need to add 2 to get 17. By going through more examples establish that the rule for the nth term of the sequence formed by the number of dots is: 3n + 2. Give a verbal justification of the rule. For example, for every shape we use 3 dots, hence the 3n. There are another two dots in the first shape, hence the + 2. Using similar arguments establish that the rule for the nth term of the sequence formed by the number of lines is: 4n + 1 and click in the corresponding cell to reveal this in the table. Click to reveal the formulae for the number of dots, d and the number of lines, l. Use similar arguments for pattern 2. Ask pupils to investigate their own patterns. Link: A4.5 Deriving formulae.

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