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© Boardworks Ltd 2004 1 of 27 A4 Sequences KS3 Mathematics

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© Boardworks Ltd 2004 2 of 27 A4.1 Introducing sequences Contents A4 Sequences A4.5 Sequences from practical contexts A4.2 Describing and continuing sequences A4.3 Generating sequences A4.4 Finding the n th term

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© Boardworks Ltd 2004 3 of 27 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 4, 8, 12, 16, 20, 24, 28, 32,... Introducing sequences

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© Boardworks Ltd 2004 4 of 27 A sequence can be infinite. That means it continues forever. For example, the sequence of multiples of 10, Infinite and finite sequences 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. For example, the sequence of two-digit square numbers 16, 25,36, 49, 64, 81 is finite.

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© Boardworks Ltd 2004 5 of 27 Some sequences follow a simple rule that is easy to describe. For example, this sequence Sequences and rules 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. In maths we are mainly concerned with sequences of numbers that follow a rule.

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© Boardworks Ltd 2004 6 of 27 Here are the names of some sequences which you may know already: 2, 4, 6, 8, 10,... 1, 3, 5, 7, 9,... 3, 6, 9, 12, 15,... 5, 10, 15, 20, 25... 1, 4, 9, 16, 25,... Even Numbers (or multiples of 2) Odd numbers Multiples of 3 Multiples of 5 Square numbers 1, 3, 6, 10,15,...Triangular numbers Naming sequences

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© Boardworks Ltd 2004 7 of 27 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, 7, 12, 17, 22, 27, 32, 37,... +5 The terms in this ascending sequence increase in unequal steps by starting at 0.1 and doubling each time. 0.1, 0.2, 0.4, 0.8, 1.6, 3.2, 6.4, 12.8,... ×2

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© Boardworks Ltd 2004 8 of 27 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, 10, 3, –4, –11, –18, –25,... –7 The terms in this descending sequence decrease in unequal steps by starting at 100 and subtracting 1, 2, 3, … 100, 99, 97, 94, 90, 85, 79, 72,... –1–2–3–4–5–6–7

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© Boardworks Ltd 2004 9 of 27 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. What other number sequences can be made from real-life situations? Sequences from real-life

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© Boardworks Ltd 2004 10 of 27 A4.2 Describing and continuing sequences Contents A4 Sequences A4.5 Sequences from practical contexts A4.1 Introducing sequences A4.3 Generating sequences A4.4 Finding the n th term

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© Boardworks Ltd 2004 11 of 27 We can show many well-known sequences using geometrical patterns of counters. Even Numbers Odd Numbers Sequences from geometrical patterns 24681013579

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© Boardworks Ltd 2004 12 of 27 Multiples of Three Multiples of Five Sequences from geometrical patterns 3691215 510152025

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© Boardworks Ltd 2004 13 of 27 Square Numbers Triangular Numbers Sequences from geometrical patterns 1 4 9 16 25 1 3 6 10 15

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© Boardworks Ltd 2004 14 of 27 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: Sequences with geometrical patterns 1 × 2 = 2 2 × 3 = 6 3 × 4 = 12 4 × 5 = 20 5 × 6 = 30

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© Boardworks Ltd 2004 15 of 27 Powers of two 2 1 = 22 2 = 4 2 3 = 82 4 = 16 2 5 = 322 6 = 64 We can show powers of two like this: Each term in this sequence is double the term before it.

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© Boardworks Ltd 2004 16 of 27 3 1 = 33 2 = 9 3 3 = 273 4 = 81 3 5 = 2433 6 = 729 Powers of three We can show powers of three like this: Each term in this sequence is three times the term before it.

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© Boardworks Ltd 2004 17 of 27 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,... +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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© Boardworks Ltd 2004 18 of 27 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 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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© Boardworks Ltd 2004 19 of 27 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: 2, 6, 8, 11, 15, 20, 26, 33,... +1+2+3+4+5+6+7 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 2004 20 of 27 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 2004 21 of 27 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,... +2+5+8+11+14+17+20 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 2004 22 of 27 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 2004 23 of 27 Can you work out the next three terms in this sequence? How did you work these out? This sequence starts with 512 and decreases by dividing by 4 each time. Sequences that decrease by dividing 512, 256, 64, 16, 4,1,0.25,0.125,... ÷4 We could also continue this sequence by multiplying by each time. 1 4

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© Boardworks Ltd 2004 24 of 27 Can you work out the next three terms in this sequence? How did you work these out? This sequence starts 1, 1 and each term is found by adding together the two previous terms. 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. 1+11+23+55+88+1313+2121+1321+34

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© Boardworks Ltd 2004 25 of 27 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 2004 26 of 27 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 2004 27 of 27 Finding missing terms

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© Boardworks Ltd 2004 28 of 27 Name that sequence!

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© Boardworks Ltd 2004 29 of 27 A4.3 Generating sequences Contents A4 Sequences A4.5 Sequences from practical contexts A4.1 Introducing sequences A4.2 Describing and continuing sequences A4.4 Finding the n th term

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© Boardworks Ltd 2004 30 of 27 Sequence grid

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© Boardworks Ltd 2004 31 of 27 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 five terms. It is finite.

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© Boardworks Ltd 2004 32 of 27 START Write down 5. Subtract 2.1. Write down the answer. Is the answer less than -5? 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 2004 33 of 27 START Write down 200. Divide by 2. Write down the answer. Is the answer less than 4? No Yes STOP Generating sequences from flow charts This flow chart generates the sequence 200, 100, 50, 25, 12.5, 6.25.

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© Boardworks Ltd 2004 34 of 27 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.

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© Boardworks Ltd 2004 35 of 27 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 2004 36 of 27 Here are some different ways in which the sequence might continue: 1 +1 2 +2 4 +3 7 +4 11 +5 16 +6 22 1 ×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 2004 37 of 27 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 1 +3 4 7 10 +3 13 +3 16 +3 19

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© Boardworks Ltd 2004 38 of 27 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 2004 39 of 27 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 sequence, 5 +2 7 +4 11 +6 17 +10 27 +12 39 +14 53...

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© Boardworks Ltd 2004 40 of 27 Write the first five terms of each sequence given the first term and the term-to-term rule. 1 st termTerm-to-term rule 21 80 48 50000 –1 1.21.2 Sequences from a term-to-term rule 10 100 3 5 7 0.8 Add 3 Subtract 5 Double Multiply by 10 Subtract 2 Add 0.1 10,13,16,19, 100,95,90,85, 3,6,12,24, 5,50,500,5000, 7,5,3,1, 0.8,0.9,1.0,1.0,1.1,1.1,

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© Boardworks Ltd 2004 41 of 27 Sometimes sequences are arranged in a table like this: Position1 st 2 nd 3 rd 4 th 5 th 6 th … n th Term369121518…3n3n 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 terms position in the sequence. What is the 100 th term in this sequence?3 × 100 = 300

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© Boardworks Ltd 2004 42 of 27 Sequences from position-to-term rules

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© Boardworks Ltd 2004 43 of 27 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),... we call the n th term T( n ). T( n ) is called the the n th term or the general term.

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© Boardworks Ltd 2004 44 of 27 For example, suppose the n th term of a sequence is 4 n + 1. We can write this rule as: T( n ) = 4 n + 1 Find the first 5 terms. T(1) =4 × 1 + 1 =5 T(2) =4 × 2 + 1 =9 T(3) =4 × 3 + 1 =13 T(4) =4 × 4 + 1 =17 T(5) =4 × 5 + 1 =21 The first 5 terms in the sequence are: 5, 9, 13, 17 and 21. Writing sequences from position-to-term rules

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© Boardworks Ltd 2004 45 of 27 If the n th term of a sequence is 2 n 2 + 3. We can write this rule as: T( n ) = 2 n 2 + 3 Find the first 4 terms. T(1) =2 × 1 2 + 3 =5 T(2) =2 × 2 2 + 3 =11 T(3) =2 × 3 2 + 3 =21 T(4) =2 × 4 2 + 3 =35 The first 4 terms in the sequence are: 5, 11, 21, and 35. Writing sequences from position-to-term rules This sequence is a quadratic sequence.

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© Boardworks Ltd 2004 46 of 27 Sequence generator – linear sequences

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© Boardworks Ltd 2004 47 of 27 Sequence generator – non-linear sequences

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© Boardworks Ltd 2004 48 of 27 Which rule is best? The term-to-term rule? The position-to-term rule? Sequences and rules

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© Boardworks Ltd 2004 49 of 27 A4.4 Finding the n th term Contents A4 Sequences A4.5 Sequences from practical contexts A4.1 Introducing sequences A4.2 Describing and continuing sequences A4.3 Generating sequences

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© Boardworks Ltd 2004 50 of 27 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 1515 2 10 3 15 4 20 5 25 n ………… × 5 5n5n

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© Boardworks Ltd 2004 51 of 27 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 1313 2626 3939 4 12 5 15 n ………… ×3 3n3n The n th term of a sequence of multiples is always dn, where d is the difference between consecutive terms.

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© Boardworks Ltd 2004 52 of 27 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 2004 53 of 27 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 3 1 2345n … × 3 3n3n Term47101316 … 3691215 + 1 3 n + 1 Each term is one more than a multiple of 3.

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© Boardworks Ltd 2004 54 of 27 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 5 1 2345n … × 5 5n5n Term16111621 … 510152025 – 4 5 n – 4 Each term is four less than a multiple of 5.

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© Boardworks Ltd 2004 55 of 27 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 –2 1 2345n … × –2 –2 n Term531–1–3 … –2–4–6–8–10 + 7 7 – 2 n Each term is seven more than a multiple of –2.

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© Boardworks Ltd 2004 56 of 27 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. For example, if an arithmetic sequence has a = 5 and d = -2, We have the sequence: 5,3,1,-1,-3,-5,... Arithmetic sequences

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© Boardworks Ltd 2004 57 of 27 The rule for the n th 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 ) = 2 n + 1 Generates odd numbers starting at 3. T( n ) = 2 n + 4 Generates even numbers starting at 6. T( n ) = 2 n – 4 Generates even numbers starting at –2. T( n ) = 3 n + 6 Generates multiples of 3 starting at 9. T( n ) = 4 – n Generates descending integers starting at 3. The n th term of an arithmetic sequence

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© Boardworks Ltd 2004 58 of 27 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 n th term

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© Boardworks Ltd 2004 59 of 27 The following sequence of patterns is made from L-shaped tiles: Number of Tiles 4812 The number of tiles in each pattern form 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 2004 60 of 27 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 2004 61 of 27 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 2004 62 of 27 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 nth 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 2004 63 of 27 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: a) Pattern 10?(3 × 10) + 1 = b) Pattern 25?(3 × 25) + 1 = c) Pattern 52?(3 × 52) + 1 = 301 31 76 156 Sequences from practical contexts

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© Boardworks Ltd 2004 64 of 27 Paving slabs 1

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© Boardworks Ltd 2004 65 of 27 Paving slabs 2

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© Boardworks Ltd 2004 66 of 27 The number of blue tiles form the sequence 8, 13, 18, 32,... Pattern number 1 Number of blue tiles 8 2 13 3 18 The rule for the n th term of this sequence is T( n ) = 5 n + 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. Paving slabs 2

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© Boardworks Ltd 2004 67 of 27 Dotty pattern 1

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© Boardworks Ltd 2004 68 of 27 Dotty pattern 2

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© Boardworks Ltd 2004 69 of 27 Leapfrog investigation

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