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**KS3 Mathematics A4 Sequences**

The aim of this unit is to teach pupils to: Generate and describe sequences Generate terms of a sequence using term-to-term and position-to-term definitions of the sequence, on paper and using ICT Find the nth term, justifying its form by referring to the context in which it was generated Material in this unit is linked to the Key Stage 3 Framework supplement of examples pp A4 Sequences

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**A4.1 Introducing sequences**

Contents A4 Sequences A4.1 Introducing sequences A4.2 Describing and continuing sequences A4.3 Generating sequences A4.4 Finding the nth term A4.5 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. Pupils may ask the difference between a sequence of numbers and a number pattern. A sequence, unlike a number pattern, does not need to follow a rule or pattern. It can follow an irregular pattern, affected by different factors (e.g. the maximum temperature each day); or consist of a random set of numbers (e.g. numbers in the lottery draw). These ideas can be discussed in more detail during the plenary session at the end of the lesson. When we write out sequences, consecutive terms are usually separated by commas.

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**Infinite and finite sequences**

A sequence can be infinite. That means it continues forever. For example, the sequence of multiples of 10, 10, 20 ,30, 40, 50, 60, 70, 80, 90 . . . is infinite. We show this by adding three dots at the end. If a sequence has a fixed number of terms it is called a finite sequence. Pupils may ask the difference between a sequence of numbers and a number pattern. A sequence, unlike a number pattern, does not need to follow a rule or pattern. It can follow an irregular pattern, affected by different factors (e.g. the maximum temperature each day); Or consist of a random set of numbers (e.g. numbers in the lottery draw). These ideas can be discussed in more detail during the plenary session at the end of the lesson. For example, the sequence of two-digit square numbers 16, 25 ,36, 49, 64, 81 is finite.

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Sequences and rules Some sequences follow a simple rule that is easy to describe. For example, this sequence 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, … continues by adding 3 each time. Each number in this sequence is one less than a multiple of three. Other sequences are completely random. For example, the sequence of winning raffle tickets in a prize draw. Ask pupils if they can see how the sequence 2, 5, 8, 11, 14, 17 … continues before revealing the solution. In maths we are mainly concerned with sequences of numbers that follow a rule.

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Naming sequences Here are the names of some sequences which you may know already: 2, 4, 6, 8, 10, . . . Even Numbers (or multiples of 2) 1, 3, 5, 7, 9, . . . Odd numbers 3, 6, 9, 12, 15, . . . Multiples of 3 5, 10, 15, 20, Multiples of 5 This can be done as an oral activity. It may also be useful for pupils to copy these number patterns into their books. As each number sequence is revealed ask the name of the sequence before revealing it. Year 7 pupils should have met square numbers and triangular numbers in Year 6. They will be revisited in more detail later in the year. Pupils may describe the pattern of square numbers either as adding 3, adding 5, adding 7 etc. (i.e. adding consecutive odd numbers) or as 1 × 1, 2 × 2, 3 × 3, 4 × 4, 5 × 5 etc. Pupils may need help verbalizing a rule to generate triangular numbers (i.e. add together consecutive whole numbers). 1, 4, 9, 16, 25, . . . Square numbers 1, 3, 6, 10,15, . . . Triangular numbers

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Ascending sequences When each term in a sequence is bigger than the one before the sequence is called an ascending sequence. For example, The terms in this ascending sequence increase in equal steps by adding 5 each time. 2, , , , , , , 37, . . . +5 +5 +5 +5 +5 +5 +5 The terms in this ascending sequence increase in unequal steps by starting at 0.1 and doubling each time. Stress the difference between sequences that increase in equal steps (linear sequences) and sequences that increase in unequal steps (non-linear) sequences. 0.1, 0.2, , , , , , , . . . ×2 ×2 ×2 ×2 ×2 ×2 ×2

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Descending sequences When each term in a sequence is smaller than the one before the sequence is called a descending sequence. For example, The terms in this descending sequence decrease in equal steps by starting at 24 and subtracting 7 each time. 24, 17, , , –4, –11, –18, –25, . . . –7 –7 –7 –7 –7 –7 –7 The terms in this descending sequence decrease in unequal steps by starting at 100 and subtracting 1, 2, 3, … Stress the difference between sequences that decrease in equal steps (linear sequences) and sequences that decrease in unequal steps (non-linear) sequences. 100, 99, , , , , , 72, . . . –1 –2 –3 –4 –5 –6 –7

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**Sequences from real-life**

Number sequences are all around us. Some sequences, like the ones we have looked at today follow a simple rule. Some sequences follow more complex rules, for example, the time the sun sets each day. Some sequences are completely random, like the sequence of numbers drawn in the lottery. Discuss examples of sequences in real life. Use this time to discuss examples of sequences which are infinite and finite and sequences which random or irregular. What other number sequences can be made from real-life situations?

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**A4.2 Describing and continuing sequences**

Contents A4 Sequences A4.1 Introducing sequences A4.2 Describing and continuing sequences A4.3 Generating sequences A4.4 Finding the nth term A4.5 Sequences from practical contexts

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**Sequences from geometrical patterns**

We can show many well-known sequences using geometrical patterns of counters. Even Numbers 2 4 6 8 10 Odd Numbers Before revealing the pattern of odd numbers ask the class how they think they will be arranged. Can they be matched in pairs? 1 3 5 7 9

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**Sequences from geometrical patterns**

Multiples of Three 3 6 9 12 15 Multiples of Five 5 10 15 20 25 Ask if multiples of any number can be arranged in rectangles.

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**Sequences from geometrical patterns**

Square Numbers 25 16 9 4 1 Triangular Numbers 15 Link: N4 powers and roots – square and triangular numbers 10 6 3 1

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**Sequences with geometrical patterns**

How could we arrange counters to represent the sequence 2, 6, 12, 20, 30, . . .? The numbers in this sequence can be written as: 1 × 2, 2 × 3, 3 × 4, 4 × 5, 5 × 6, . . . We can show this sequence using a sequence of rectangles: 5 × 6 = 30 4 × 5 = 20 Ask pupils to think of numbers which multiply together to make the numbers in the sequence. 2 is prime so it can only be written as 1 × 2. 6 can be written as 1 × 6 or 2 × 3 and 12 can be written as 1 × 12, 2 × 6 or 3 × 4. Ask pupils how we could we write these factors so that they form a pattern. Establish that the sequence can be written as, 1 × 2, 2 × 3, 3 × 4, 4 × 5, 5 × 6, . . . Ask, How are these rectangles increasing in size each time? Each rectangle is one column wider and one row higher than the previous one. 3 × 4 = 12 2 × 3 = 6 1 × 2 = 2

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**Powers of two We can show powers of two like this: 25 = 32 26 = 64**

23 = 8 24 = 16 21 = 2 22 = 4 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. Each term in this sequence is double the term before it.

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**Powers of three We can show powers of three like this: 35 = 243**

36 = 729 33 = 27 34 = 81 31 = 3 32 = 9 Discuss the patterns made by powers of three. Each arrangement has three times as many dots as the one before. Even powers are squares as before, odd powers are three times a square number. Do you think all powers would display a similar pattern? Do you think all even powers are square numbers? Establish that they are. Explain that 3 to the power of 4 is the same as 9 × 9 or 9 squared. (3 × 3) × (3 × 3). Tell pupils that every whole number multiplied by itself an even number of times will give a square number. Discuss this further giving more examples if required. For example, 5 to the power of 6, is 5 × 5 × 5 × 5 × 5 × 5, 6 times. We could think of this as (5 × 5 × 5) × (5 × 5 × 5) or 1252. Ask pupils to explain why these arrangement of dots are an impractical way of displaying sequences containing increasing powers. Establish that the numbers get very big, very quickly. The sixth pattern in the sequence of powers of 4, 4 to the power of 6, would contain 4096 counters or dots! Ask pupils how many dots there would be in the sixth pattern in the sequence of powers of ten (one million!). Link: N4 Powers and roots - powers Each term in this sequence is three times the term before it.

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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 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. Introduce the word difference and encourage pupils to find the difference between consecutive terms. Remind pupils that they must check that every number in the sequence obeys the same rule. Each term in the sequence is two less than a multiple of 6. 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: 2, , , , , , , , . . . +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. Stress that this sequence increases in unequal steps. It is not linear. If we looked at the differences between the consecutive terms they would be 2, 6, 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? 512, 256, 64, 16, , 1, 0.25, 0.125, ÷4 ÷4 ÷4 ÷4 ÷4 ÷4 ÷4 How did you work these out? This sequence starts with 512 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. 1 4

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**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 3+5 5+8 8+13 13+21 21+13 21+34 How did you work these out? This sequence starts 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. If the sequence is generated by two negative numbers then it will be a descending sequence. 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 type of sequence 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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Name that sequence! Use this activity to revise the names of some known (and perhaps unknown!) sequences.

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**A4.3 Generating sequences**

Contents A4 Sequences A4.1 Introducing sequences A4.2 Describing and continuing sequences A4.3 Generating sequences A4.4 Finding the nth term A4.5 Sequences from practical contexts

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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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**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. Add on 1.5. Write down the answer. This sequence has only five 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 -5? No Go through this flow chart in the same way. The sequence generated is 5, 2.9, 0.8, -1.3, -3.4. Yes STOP

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**Generating sequences from flow charts**

START Write down 200. This flow chart generates the sequence 200, 100, 50, 25, 12.5, 6.25. Divide by 2. Write down the answer. Is the answer less than 4? No Go through this flow chart in the same way. The sequence generated is 200, 100, 50, 25, 12.5, 6.25. 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. Add together the two previous numbers. Write down the answer. Is the answer more than 100? Go through this flow chart in the same way. The sequence generated is 3, 4, 7, 11, 18, 29, 47, 76. 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. Is it possible to find one for which: a) all the numbers are multiples of 3? b) all the numbers are odd? c) all the numbers are multiples of 9? d) none of the numbers is a whole number? 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 sequence, 5 7 11 17 27 39 53 . . . +2 +4 +6 +10 +12 +14

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**Sequences from a term-to-term rule**

Write the first five terms of each sequence given the first term and the term-to-term rule. 1st term Term-to-term rule 10 Add 3 10, 13, 16, 19, 21 100 Subtract 5 100, 95, 90, 85, 80 3 Double 3, 6, 12, 24, 48 Edit the numbers in this slide to produce more or less challenging examples. 5 Multiply by 10 5, 50, 500, 5000, 50000 7 Subtract 2 7, 5, 3, 1, –1 0.8 Add 0.1 0.8, 0.9, 1.0, 1.1, 1.2

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

Sometimes sequences are arranged in a table like this: Position 1st 2nd 3rd 4th 5th 6th … nth Term 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 is 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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**Writing sequences from position-to-term rules**

The position-to-term rule for a sequence is very useful because it allows us to work out any term in the sequence without having to work out any other terms. We can use algebraic shorthand to do this. We call the first term T(1), for Term number 1, we call the second term T(2), we call the third term T(3), . . . Explain the algebraic notation. Reassure pupils that using letters is a good way to save us lots of writing! The T stands for term and the number in the bracket is the position of the term in the sequence. n can be any whole number. Ask, What do we call the 10th term of a sequence? (T(10)) What do we call the 450th term? And so on. Explain again that we find the value of a term by substituting its position number into the rule for the nth term. we call the nth term T(n). T(n) is called the the nth term or the general term.

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

For example, suppose the nth term of a sequence is 4n + 1. We can write this rule as: T(n) = 4n + 1 Find the first 5 terms. T(1) = 4 × = 5 T(2) = 4 × = 9 T(3) = 4 × = 13 Repeat that T(1) is short way of writing the first term. To find the value of T(1) substitute 1 into the rule 4n + 1. To find the value of T(2) substitute 2 into the rule 4n + 1 etc. What do you notice about this sequence? (It goes up 4 each time. An even better answer is: it is the numbers from the 4 times table with 1 added on each time.) Ask pupils to use the rule to work out the value of the 10th term, the 50th term, the 243rd term, etc. T(4) = 4 × = 17 T(5) = 4 × = 21 The first 5 terms in the sequence are: 5, 9, 13, 17 and 21.

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

If the nth term of a sequence is 2n2 + 3. We can write this rule as: T(n) = 2n2 + 3 Find the first 4 terms. T(1) = 2 × = 5 T(2) = 2 × = 11 T(3) = 2 × = 21 Stress that in a linear sequence the n can only be raised to the power of 1 (though this is not usually written). In a quadratic sequence n can be raised to the power of 1 or the power of 2. T(4) = 2 × = 35 The first 4 terms in the sequence are: 5, 11, 21, and 35. This sequence is a quadratic sequence.

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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 non-linear sequences. 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’. You may like to introduce the phrase ‘general term’ although this is not a key word in Year 7. 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. Note: For some sequences that increase in unequal steps, the position-to-term rule is not so useful – for example, in The Fibonacci Sequence where each term is the sum of the previous two terms.

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**A4 Sequences Contents A4.1 Introducing sequences**

A4.2 Describing and continuing sequences A4.3 Generating sequences A4.4 Finding the nth term A4.5 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, , , , , … +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: Ask pupils, Can you find a rule linking the position of the term to 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: Position 1 3 2 6 3 9 4 12 5 15 … n Stress that for any sequence of multiples the nth term will be the difference between the consecutive terms (or d) multiplied by n. ×3 ×3 ×3 ×3 ×3 ×3 Term 3n The nth term of a sequence of multiples is always dn, where d is the difference between consecutive terms.

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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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Arithmetic sequences Sequences that increase (or decrease) in equal steps are called linear or arithmetic sequences. The difference between any two consecutive terms in an arithmetic sequence is a constant number. When we describe arithmetic sequences we call the difference between consecutive terms, d. We call the first term in an arithmetic sequence, a. Explain what is meant by consecutive terms (terms that are next to each other in the sequence) and constant number (the same number each time). Reinforce verbally that the difference between consecutive terms is always the same number because we are adding on the same amount each time. We call the amount we add on each time, d, for difference. Before revealing the third sentence, ask, If we want to generate an arithmetic sequence what else do we need to know, apart from how much to add on each time? Suppose we wanted to generate a sequence that went up 4 each time. We need a start number – the first term in the sequence. For example, if an arithmetic sequence has a = 5 and d = -2, We have the sequence: 5, 3, 1, -1, -3, -5, . . .

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**The nth term of an arithmetic sequence**

The rule for the nth term of any arithmetic sequence is of the form: T(n) = an + b a and b can be any number, including fractions and negative numbers. For example, T(n) = 2n + 1 Generates odd numbers starting at 3. T(n) = 2n + 4 Generates even numbers starting at 6. For each example discuss the sequence generated by asking pupils to substitute values for n to give the first few terms of the sequence. Assure pupils that T(n) = 4 – n has the form T(n) = an + b. In this case, a is –1 and b is 4. T(n) = -n + 4 is equivalent to T(n) = 4 – n. We write it the other way round because it is ‘neater’ if there isn’t a negative sign at the beginning. Remember: -1 n is just written as –n. T(n) = 2n – 4 Generates even numbers starting at –2. T(n) = 3n + 6 Generates multiples of 3 starting at 9. T(n) = 4 – n Generates descending integers starting at 3.

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**A4.5 Sequences from practical contexts**

Contents A4 Sequences A4.1 Introducing sequences A4.2 Describing and continuing sequences A4.3 Generating sequences A4.4 Finding the nth term A4.5 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, What kind of sequence is this? This is an arithmetic sequence because the same number of tiles is added on each time. (It is also the sequence of multiples of 4). How do you know that 4 tiles will be added on to make the next pattern? 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 form 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 4n is the position-to-term rule or rule for the nth term. It can also be called the general term of the sequence. Remember n is the position of the term in the sequence. n can be any whole number bigger than 1. Another way to say this rule, then, is 4 times the position number. Whenever you give a rule for a pattern you should try to justify it to prove it works. 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: a) Pattern 10? (3 × 10) + 1 = 31 Use substitution to find terms given their pattern number. Ask 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) Pattern 25? (3 × 25) + 1 = 76 c) Pattern 52? (3 × 52) + 1 = 156

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Paving slabs 1 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, Can you 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 tile by multiplying the number of yellow tiles by 8.

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**Paving slabs 2 This is the next pattern the gardener tried.**

At first glance this may look like the first tiling pattern. Do a few random examples to illustrate the pattern, then instruct pupils to investigate the pattern further. Tell them to draw out the first three patterns of the sequence, put their results into a table and then use their results to predict the next (4th) term. Tell pupils that they must check their prediction by either drawing out the next pattern or by justifying their prediction in words by referring to the pattern. Ask pupils to find the rule for the nth term of the pattern of blue tiles and to justify it in words. Lastly pupils should use their rule to work out how many blue tiles the gardener would use if he used 100 yellow tiles (in other words, what is the 100th term in the sequence of blue tiles). A justification of the rule is shown on the next slide.

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Paving slabs 2 The number of blue tiles form the sequence 8, 13, 18, 32, . . . Pattern number 1 2 3 Number of blue tiles 8 13 18 The rule for the nth term of this sequence is T(n) = 5n + 3 Justification: Each time we add another yellow tile we add 5 blue tiles. The +3 comes from the 3 tiles at the start of each pattern.

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Dotty pattern 1 Gabby is drawing patterns on square dotty paper. For each shape in the pattern she writes down the number of dots she passes through and the number of lines connecting the dots. She puts her results in 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. After a few values have been entered ask pupils to predict the remaining values. Ask pupils how they made their predictions. 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.

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Dotty pattern 2 Gabby tries another pattern. For each shape in the pattern she write down the number of dots she passes through and the number of lines connecting the dots. She puts her results in a table. Again, 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. Establish that the rule for the nth term of the sequence formed by the number of dots is 2n + 2 and that the rule for the nth term of the sequence formed by the number of lines is 3n + 1. Ask pupils to justify these rules using the pattern.

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**Leapfrog investigation**

Explain that the object of this investigation is to swap the frogs so that the green frogs are on the left and red frogs are on the right. A frog can only move in the direction it is facing. It can slide forward one place to an empty lily pad or it can jump over a different coloured frog provided there is an empty space behind it. Start by developing a strategy to solve the puzzle. Note the number of slides and the number of jumps required. Compete the puzzle for one frog on each side, two frogs on each side and three frogs on each side. Ask pupils to predict the pattern of slides and jumps if we start with four frogs on each side. Suggest to pupils that they could call the number of frogs on each side n and find the following formulae: The number of slides in terms of n. The number of jumps in terms of n. The total number of moves in term of n. Extend the activity by using different numbers of frogs on each side and by changing the number of empty lily pads in the middle.

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