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KS3 Mathematics

KS3 Mathematics. A4 Sequences. A4 Sequences. Contents. A4.2 Describing and continuing sequences. A4.1 Introducing sequences. A4.3 Generating sequences. A4.4 Finding the n th term. A4.5 Sequences from practical contexts. Introducing sequences. 4, 8, 12, 16, 20, 24, 28, 32,. 1 st term.

robert-larson
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KS3 Mathematics

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  1. KS3 Mathematics A4 Sequences

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

  3. Introducing sequences 4, 8, 12, 16, 20, 24, 28, 32, . . . 1st term 6th term 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.

  4. 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. For example, the sequence of two-digit square numbers 16, 25 ,36, 49, 64, 81 is finite.

  5. 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. In maths we are mainly concerned with sequences of numbers that follow a rule.

  6. 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, 25 . . . Multiples of 5 1, 4, 9, 16, 25, . . . Square numbers 1, 3, 6, 10,15, . . . Triangular numbers

  7. Ascending sequences ×2 +5 +5 +5 +5 +5 +5 +5 ×2 ×2 ×2 ×2 ×2 ×2 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, . . . 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, . . .

  8. Descending sequences –7 –7 –7 –7 –7 –7 –7 –7 –1 –2 –3 –4 –5 –6 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, . . . 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, . . .

  9. Sequences from real-life Some sequences are completely random, like the sequence of numbers drawn in the lottery. 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. What other number sequences can be made from real-life situations?

  10. 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

  11. Sequences from geometrical patterns 2 4 6 8 10 1 3 5 7 9 We can show many well-known sequences using geometrical patterns of counters. Even Numbers Odd Numbers

  12. Sequences from geometrical patterns 3 6 9 12 15 5 10 15 20 25 Multiples of Three Multiples of Five

  13. Sequences from geometrical patterns 4 1 9 16 25 1 3 6 10 15 Square Numbers Triangular Numbers

  14. Sequences with geometrical patterns 2 × 3 = 6 3 × 4 = 12 4 × 5 = 20 5 × 6 = 30 1 × 2 = 2 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:

  15. Powers of two 22 = 4 23 = 8 21 = 2 24 = 16 25 = 32 26 = 64 We can show powers of two like this: Each term in this sequence is double the term before it.

  16. Powers of three 31 = 3 32 = 9 33 = 27 34 = 81 35 = 243 36 = 729 We can show powers of three like this: Each term in this sequence is three times the term before it.

  17. Sequences that increase in equal steps +4 +4 +4 +4 +4 +4 +4 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.

  18. Sequences that decrease in equal steps –6 –6 –6 –6 –6 –6 –6 Can you work out the next three terms in this sequence? 22, 16, 10, 4, –2, –8, –14, –20, . . . How did you work these out? This sequence starts with 22 and decreases by 6 each time. 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.

  19. Sequences that increase in increasing steps +1 +2 +3 +4 +5 +6 +7 Some sequences increase or decrease in unequal steps. For example, look at the differences between terms in this sequence: 2, 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.

  20. Sequences that decrease in decreasing steps –0.1 –0.2 –0.3 –0.4 –0.5 –0.6 –0.7 Can you work out the next three terms in this sequence? 7, 6.9, 6.7, 6.4, 6, 5.5, 4.9, 4.2, . . . How did you work these out? This sequence starts with 7 and decreases by 0.1, 0.2, 0.3, 0.4, 0.5, … With sequences of this type it is often helpful to find a second row of differences.

  21. Using a second row of differences +20 +17 +14 +8 +5 +2 +3 +3 +3 +3 +3 +11 +3 Can you work out the next three terms in this sequence? 1, 3, 8, 16, 27, 41, 58, 78, . . . 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.

  22. Sequences that increase by multiplying ×2 ×2 ×2 ×2 ×2 ×2 ×2 Some sequences increase or decrease by multiplying or dividing each term by a constant factor. For example, look at this sequence: 2, 4, 8, 16, 32, 64, 128, 256, . . . 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.

  23. Sequences that decrease by dividing ÷4 ÷4 ÷4 ÷4 ÷4 ÷4 ÷4 We could also continue this sequence by multiplying by each time. 1 4 Can you work out the next three terms in this sequence? 512, 256, 64, 16, 4, 1, 0.25, 0.125, . . . How did you work these out? This sequence starts with 512 and decreases by dividing by 4 each time.

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

  25. 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 calledlinearorarithmetic 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. • Sequences that increase or decrease by adding together • the two previous terms.

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

  27. Finding missing terms

  28. Name that sequence!

  29. 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

  30. Sequence grid

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

  32. Generating sequences from flow charts No Yes 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? STOP

  33. Generating sequences from flow charts No Yes 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? STOP

  34. Generating sequences from flow charts No Yes 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? STOP

  35. Predicting terms in a sequence For example, 87, 84, 81, 78, . . . Usually, we can predict how a sequence will continue by looking for patterns. 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?

  36. Continuing sequences +1 +2 +3 +4 +5 +6 ×2 ×2 ×2 ×2 ×2 ×2 Here are some different ways in which the sequence might continue: 1 2 4 7 11 16 22 1 2 4 8 16 32 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.

  37. Continuing sequences +3 +3 +3 +3 +3 +3 This sequence continues by adding 3 each time. 1 4 7 10 13 16 19 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.

  38. 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, . . . 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, . . .

  39. Writing sequences from term-to-term-rules +2 +4 +6 +10 +12 +14 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 . . .

  40. 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 Subtract 5 100, 95, 90, 85, 80 100 3 Double 3, 6, 12, 24, 48 Multiply by 10 5, 50, 500, 5000, 50000 5 7 Subtract 2 7, 5, 3, 1, –1 0.8 Add 0.1 0.8, 0.9, 1.0, 1.1, 1.2

  41. Sequences from position-to-term rules Sometimes sequences are arranged in a table like this: 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 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

  42. Sequences from position-to-term rules

  43. 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 nth term T(n). T(n) is called the the nth term or the general term.

  44. 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 ×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.

  45. 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 ×12 + 3 = 5 T(2) = 2 ×22 + 3 = 11 T(3) = 2 ×32 + 3 = 21 T(4) = 2 ×42 + 3 = 35 The first 4 terms in the sequence are: 5, 11, 21, and 35. This sequence is a quadratic sequence.

  46. Sequence generator – linear sequences

  47. Sequence generator – non-linear sequences

  48. Sequences and rules Which rule is best? The term-to-term rule? The position-to-term rule?

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

  50. Sequences of multiples +5 +5 +5 +5 +5 +5 +5 × 5 × 5 × 5 × 5 × 5 × 5 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 … can be found by adding 5 each time. Compare the terms in the sequence of multiples of 5 to their position in the sequence: 2 10 3 15 4 20 5 25 n Position 1 5 … … 5n Term

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