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© Boardworks Ltd of 62 A3 Sequences S1–S4 Mathematics

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© Boardworks Ltd of 62 A A A A A A3.1 Special number patterns A3 Sequences Contents A3.2 Describing and continuing sequences A3.3 Generating sequences from rules A3.5 Sequences from practical contexts A3.4 Finding the n th term of a linear sequence

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© Boardworks Ltd of 62 Even numbers Even numbers are numbers that are exactly divisible by 2. All even numbers end in 0, 2, 4, 6 or 8. For example, 48 is an even number. It can be written as 48 = 2 × 24. Even numbers can be illustrated using dots or counters arranged as follows:

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© Boardworks Ltd of 62 Odd numbers Odd numbers leave a remainder of 1 when divided by 2. All odd numbers end in 1, 3, 5, 7 or 9. For example, 17 is an odd number. It can be written as 17 = 2 × Odd numbers can be illustrated using dots or counters arranged as follows: 13579

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© Boardworks Ltd of 62 Making triangles

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© Boardworks Ltd of 62 Triangular numbers Triangular numbers are numbers that can be written as the sum of consecutive whole numbers starting with 1. Triangular numbers can be illustrated using dots or counters arranged in triangles: For example, 15 is a triangular number. It can be written as 15 =

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© Boardworks Ltd of 62 Making squares

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© Boardworks Ltd of 62 Square numbers Square numbers are obtained when a whole number is multiplied by itself. They are sometimes called perfect squares. Square numbers can be illustrated using dots or counters arranged in squares: For example, 49 is a square number. It can be written as 49 = 7 × 7 or 49 =

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© Boardworks Ltd of 62 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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© Boardworks Ltd of 62 Adding consecutive odd numbers The first square number is 1.The second square number is 4.The third square number is 9.The fourth square number is 16.The fifth square number is 25.The sixth square number is 36.The seventh square number is 49.The eighth square number is 64.The ninth square number is 81.The tenth square number is = = = = = = = = = 100

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© Boardworks Ltd of 62 We can also make square numbers by adding two consecutive triangular numbers = = = = = = = = = 100 Adding two consecutive triangular numbers

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© Boardworks Ltd of 62 Powers of = 22 2 = = 82 4 = = = 64 We can show powers of two like this: Each number is double the one before it. Powers of 2 are found by multiplying the number 2 by itself a given number of times. For example, 2 × 2 × 2 = 2 3 = 8.

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© Boardworks Ltd of 62 Cube numbers Cube numbers are obtained when a whole number is multiplied by itself and then by itself again. Cube numbers can be illustrated using spheres arranged in cubes: For example, 64 is a cube number. It can be written as 64 = 4 × 4 × 4 or 64 =

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© Boardworks Ltd of 62 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:

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© Boardworks Ltd of 62 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:

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© Boardworks Ltd of 62 Name this sequence

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© Boardworks Ltd of 62 Squares, triangles and primes

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© Boardworks Ltd of 62 A A A A A Contents A3.2 Describing and continuing sequences A3 Sequences A3.1 Introducing sequences A3.3 Generating sequences from rules A3.5 Sequences from practical contexts A3.4 Finding the n th term of a linear sequence

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© Boardworks Ltd of 62 4, 8, 12, 16, 20, 24, 28, 32,... In maths, we call a list of numbers in order a sequence. Each number in a sequence is called a 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. 1 st term6 th term Introducing sequences

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© Boardworks Ltd of 62 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, 7, 11, 15, 19, 23, 27, 31, 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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© Boardworks Ltd of 62 Can you work out the next three terms in this sequence? How did you work these out? This sequence starts with 22 and decreases by 6 each time. Sequences that decrease in equal steps 22, 16, 10, 4, –2,–8,–14,–20,... –6 Sequences that increase or decrease in equal steps are called linear or arithmetic sequences.

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© Boardworks Ltd of 62 Some sequences increase or decrease in unequal steps. Sequences that increase in increasing steps For example, look at the differences between terms in this sequence: 5, 6, 8, 11, 15, 20, 26, 33, 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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© Boardworks Ltd of 62 Can you work out the next three terms in this sequence? How did you work these out? This sequence starts with 7 and decreases by 0.1, 0.2, 0.3, 0.4, 0.5, … Sequences that decrease in decreasing steps 7, 6.9, 6.7, 6.4, 6,5.5,4.9,4.2,... –0.1–0.2–0.3–0.4–0.5–0.6–0.7 With sequences of this type it is often helpful to find a second row of differences.

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© Boardworks Ltd of 62 Can you work out the next three terms in this sequence? Look at the differences between terms. Using a second row of differences 1, 3, 8, 16, 27,41,58,78, A sequence is formed by the differences so we look at the second row of differences. +3 This shows that the differences increase by 3 each time.

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© Boardworks Ltd of 62 Some sequences increase or decrease by multiplying or dividing each term by a constant factor. Sequences that increase by multiplying For example, look at this sequence: 2, 4, 8, 16, 32, 64, 128, 256,... ×2 This sequence starts with 2 and increases by multiplying the previous term by 2. All of the terms in this sequence are powers of 2.

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© Boardworks Ltd of 62 Can you work out the next three terms in this sequence? How did you work these out? This sequence starts with 1024 and decreases by dividing by 4 each time. Sequences that decrease by dividing We could also continue this sequence by multiplying by each time, since that is equivalent to dividing by , 256, 64, 16, 4,1, ÷4, ,

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© Boardworks Ltd of 62 Can you work out the next three terms in this sequence? How did you work these out? This sequence starts with 1, 1 and each term is found by adding together the two previous terms. Fibonacci and Fibonacci-type sequences 1, 1, 2, 3, 5, 8, 13,21,34,55,... This sequence is called the Fibonacci sequence after the Italian mathematician who first wrote about it

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© Boardworks Ltd of 62 Describing and continuing sequences Here are some of the types of sequence you may come across: Sequences that increase or decrease by adding together the two previous terms. 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.

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© Boardworks Ltd of 62 A number sequence starts as follows 1, 2,... Continuing sequences How many ways can you think of continuing the sequence? Give the next three terms and the rule for each one.

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© Boardworks Ltd of 62 Finding missing terms

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© Boardworks Ltd of 62 Sequence grid

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© Boardworks Ltd of 62 Contents A A A A A A3.3 Generating sequences from rules A3 Sequences A3.2 Describing and continuing sequences A3.1 Introducing sequences A3.5 Sequences from practical contexts A3.4 Finding the n th term of a linear sequence

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© Boardworks Ltd of 62 Generating sequences from flow charts A sequence can be given by a flow chart. For example, START Write down 3. Add on 1.5. Write down the answer. Is the answer more than 10? No Yes STOP This flow chart generates the sequence: 3, 4.5, 6, 6.5, 9, This sequence has only six terms. It is finite.

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© Boardworks Ltd of 62 START Write down 5. Subtract 2.1. Write down the answer. Is the answer less than –3? No Yes STOP Generating sequences from flow charts This flow chart generates the sequence 5, 2.9, 0.8, –1.3, –3.4.

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© Boardworks Ltd of 62 START Write down 3 and 4. Add together the two previous numbers. Write down the answer. Is the answer more than 100? No Yes STOP Generating sequences from flow charts This flow chart generates the sequence 3, 4, 7, 11, 18, 29, 47, 76, 123.

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© Boardworks Ltd of 62 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. For example, A sequence starts with the numbers 1, 2, 4,... How could this sequence continue?

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© Boardworks Ltd of 62 Here are some different ways in which the sequence might continue: ×2×2 2 ×2×2 4 ×2×2 8 ×2×2 16 ×2×2 32 ×2×2 64 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. Continuing sequences

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© Boardworks Ltd of 62 This sequence continues by adding 3 each time. 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. Continuing sequences

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© Boardworks Ltd of 62 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 2,17,20,23,...5,8,11,14, If we start with 0.4 and add on 3 each time we have 0.4,15.4,18.4,21.4,...3.4,6.4,9.4,12.4, Using a term-to-term rule

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© Boardworks Ltd of 62 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, 1 st term 5 Term-to-term rule Add consecutive even numbers starting with 2. This gives us the following sequence:

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© Boardworks Ltd of 62 Write the first five terms of each sequence given the first term and the term-to-term rule. 1 st termTerm-to-term rule – Sequences from term-to-term rules –7Add 3 100Subtract 8 3Double 0.04 Multiply by Subtract Divide by 5 –7,–4,–1,2, 100,92,84,76, 3,6,12,24, 0.04,0.4,4,40, 1.3,0.7,0.1,–0.5, 111,22.2,4.44,0.888, First five terms

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© Boardworks Ltd of 62 Sometimes sequences are arranged in a table like this: Term n th …6 th 5 th 4 th 3 rd 2 nd 1 st Position Sequences from position-to-term rules 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. For this sequence we can say that the n th term is 3 n, where n is a term’s position in the sequence. What is the 100 th term in this sequence? 3 × 100 = 300 3n3n …

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© Boardworks Ltd of 62 Sequences from position-to-term rules

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© Boardworks Ltd of 62 Sequence generator – linear sequences

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© Boardworks Ltd of 62 Sequence generator – non-linear sequences

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© Boardworks Ltd of 62 Which rule is best? The term-to-term rule? The position-to-term rule? Sequences and rules

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© Boardworks Ltd of 62 Contents A A A A A A3.4 Finding the n th term of a linear sequence A3 Sequences A3.3 Generating sequences from rules A3.2 Describing and continuing sequences A3.1 Introducing sequences A3.5 Sequences from practical contexts

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© Boardworks Ltd of 62 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, 10, 15, 20, 25, 30, 35, 40, … +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: Position Term n ………… × 5 5n5n

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© Boardworks Ltd of 62 Sequences of multiples The sequence of multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, … +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 Term n ………… ×3 3n3n

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© Boardworks Ltd of 62 Sequences of multiples The n th term of a sequence of multiples is always dn, where d is the difference between consecutive terms. For example, The n th term of 4, 8, 12, 16, 20, 24, … is4n4n The 10 th term of this sequence is 4 × 10 = 40 The 25 th term of this sequence is 4 × 25 = 100 The 47 th term of this sequence is 4 × 47 = 188

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© Boardworks Ltd of 62 Finding the n th term of a linear sequence The terms in this sequence 4, 7, 10, 13, 16, 19, 22, 25, … +3 can be found by adding 3 each time. Compare the terms in the sequence to the multiples of 3. Position Multiples of n … × 3 3n3n Term … n + 1 Each term is one more than a multiple of 3.

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© Boardworks Ltd of 62 Finding the n th term of a linear sequence The terms in this sequence 1, 6, 11, 16, 21, 26, 31, 36, … +5 can be found by adding 5 each time. Compare the terms in the sequence to the multiples of 5. Position Multiples of n … × 5 5n5n Term … – 4 5 n – 4 Each term is four less than a multiple of 5.

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© Boardworks Ltd of 62 Finding the n th term of a linear sequence The terms in this sequence 5, 3, 1, –1, –3, –5, –7, –9, … –2 can be found by subtracting 2 each time. Compare the terms in the sequence to the multiples of –2. Position Multiples of – n … × –2 –2 n Term531–1–3 … –2–4–6–8– – 2 n Each term is seven more than a multiple of –2.

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© Boardworks Ltd of 62 Finding the n th term of a linear sequence

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© Boardworks Ltd of 62 Contents A A A A A A3.5 Sequences from practical contexts A3 Sequences A3.3 Generating sequences from rules A3.2 Describing and continuing sequences A3.1 Introducing sequences A3.4 Finding the n th term of a linear sequence

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© Boardworks Ltd of 62 The following sequence of patterns is made from L-shaped tiles: Number of Tiles 4812 The number of tiles in each pattern forms a sequence. How many tiles will be needed for the next pattern? 16 We add on four tiles each time.This is a term-to-term rule. Sequences from practical contexts

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© Boardworks Ltd of 62 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 42 lots of 43 lots of 44 lots of 4 The n th term is 4 × n or 4 n. Justification: This follows because the 10 th term would be 10 lots of 4. Sequences from practical contexts

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© Boardworks Ltd of 62 Now, look at this pattern of blocks: Number of Blocks 4710 How many blocks will there be in the next shape? 13 We add on 3 blocks each time. This is the term-to term rule. Justification: The shapes have three ‘arms’ each increasing by one block each time. Sequences from practical contexts

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© Boardworks Ltd of 62 How many blocks will there be in the 100 th arrangement? We need a rule for the n th term. Look at pattern again: 1 st pattern2 nd pattern3 rd pattern4 th pattern The n th pattern has 3 n + 1 blocks in it. Justification: The patterns have 3 ‘arms’ each increasing by one block each time. So the n th pattern has 3 n blocks in the arms, plus one more in the centre. Sequences from practical contexts

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© Boardworks Ltd of 62 So, how many blocks will there be in the 100 th pattern? Number of blocks in the n th pattern = 3 n + 1 When n is 100, Number of blocks =(3 × 100) + 1 = How many blocks will there be in the: a) 10 th pattern?(3 × 10) + 1 = b) 25 th pattern?(3 × 25) + 1 = c) 52 nd pattern?(3 × 52) + 1 = Sequences from practical contexts

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© Boardworks Ltd of 62 Paving slab patterns

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© Boardworks Ltd of 62 Dotty patterns

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