12 Number PatternsMEP Pupil Text 12 12.1 Simple Number Patterns A list of numbers which form a pattern is called a sequence. In this section, straightfor- ward sequences are continued. Worked Example 1 Write down the next three numbers in each sequence. (a) 2, 4, 6, 8, 10, . (b) 3, 6, 9, 12, 15, . Solution (a) This sequence is a list of even numbers, so the next three numbers will be 12, 14, 16. (b) This sequence is made up of the multiples of 3, so the next three numbers will be 18, 21, 24. Worked Example 2 Find the next two numbers in each sequence. (a) 6, 10, 14, 18, 22, . (b) 3, 8, 13, 18, 23, . Solution (a) For this sequence the difference between each term and the next term is 4. Sequence 6, 10, 14, 18, 22, . Difference 4 4 4 4 So 4 must be added to obtain the next term in the sequence. The next two terms are 22+= 4 26 and 26+= 4 30 , giving 6, 10, 14, 18, 22, 26, 30, . (b) For this sequence, the difference between each term and the next is 5. Sequence 3, 8, 13, 18, 23, . Difference 5 5 5 5 Adding 5 gives the next two terms as 23+= 5 28 and 28+= 5 33, giving 3, 8, 13, 18, 23, 28, 33, . 262 MEP Pupil Text 12 Exercises 1. Write down the next four numbers in each list. (a) 1, 3, 5, 7, 9, . (b) 4, 8, 12, 16, 20, . (c) 5, 10, 15, 20, 25, . (d) 7, 14, 21, 28, 35, . (e) 9, 18, 27, 36, 45, . (f) 6, 12, 18, 24, 30, . (g) 10, 20, 30, 40, 50, . (h) 11, 22, 33, 44, 55, . (i) 8, 16, 24, 32, 40, . (j) 20, 40, 60, 80, 100, . (k) 15, 30, 45, 60, 75, . (l) 50, 100, 150, 200, 250, . 2. Find the difference between terms for each sequence and hence write down the next two terms of the sequence. (a) 5, 8, 11, 14, 17, . (b) 2, 10, 18, 26, 34, . (c) 7, 12, 17, 22, 27, . (d) 6, 17, 28, 39, 50, . 1 1 1 (e) 8, 15, 22, 29, 36, . (f) 2 2 , 3, 3 2 , 4, 4 2 , . (g) 4, 13, 22, 31, 40, . (h) 26, 23, 20, 17, 14, . (i) 20, 16, 12, 8, 4, . (j) 18, 14, 10, 6, 2, . (k) 11, 8, 5, 2, −1, . (l) − 5, −8, −11, −14 , −17, . 3. In each part, find the answers to (i) to (iv) with a calculator and the answer to (v) without a calculator. (a) (i) 211×=? (b) (i) 99×= 11 ? (ii) 22×= 11 ? (ii) 999×= 11 ? (iii) 222×= 11 ? (iii) 9999×= 11 ? (iv) 2222×= 11 ? (iv) 99999×= 11 ? (v) 22222×= 11 ? (v) 999999×= 11 ? (c) (i) 88×= 11 ? (d) (i) 79×=? (ii) 888×= 11 ? (ii) 799×=? (iii) 8888×= 11 ? (iii) 7×= 999 ? (iv) 88888×= 11 ? (iv) 7×= 9999 ? (v) 8888888×= 11 ? (v) 7×= 999999 ? 4. (a) (i) Complete the following number pattern: 11 = 11 11× 11 = 121 11×× 11 11 = ? ? = ? (ii) Look at the numbers in the right hand column. Write down what you notice about these numbers. 263 12.1 MEP Pupil Text 12 (b) Use your calculator to work out the next line of the pattern. What do you notice now? (NEAB) 5. 92×= 18 93×= 27 94×= 36 95×= 45 96×= 54 97×= 63 98×= 72 99×= 81 (a) (i) Complete the statement, The units digits in the right hand column, in order, are 8, 7, 6, . (ii) Complete the statement, The tens digits in the right-hand column, in order, are 1, 2, 3, . (iii) What is the connection between the answers to parts (i) and (ii)? (b) The numbers in the right-hand column go up by 9 each time. What else do you notice about these numbers? (MEG) 12.2 Recognising Number Patterns This section looks at how the terms of a sequence are related. For example the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, . is obtained by adding together two consecutive terms to obtain the next term. 11+= 2 12+= 3 23+= 5 35+= 8 58+= 13 Worked Example 1 The sequence of square numbers is 1, 4, 9, 16, 25, 36, . Explain how to obtain the next number in the sequence. Solution The next number can be obtained in one of two ways. 264 MEP Pupil Text 12 (1) The sequence could be written, 12 , 22 , 32 , 42 , 52 , 62 , . and so the next term would be 7492=. (2) Calculating the differences between the terms gives Sequence 1, 4, 9, 16, 25, 36, . Difference 3 5 7 9 11 The difference increases by 2 each time so the next term would be 36+= 13 49. Worked Example 2 Describe how to obtain the next term of each sequence below. (a) 3, 10, 17, 24, 31, . (b) 3, 6, 11, 18, 27, . (c) 1, 5, 6, 11, 17, 28, . Solution (a) Finding the differences between the terms gives Sequence 3, 10, 17, 24, 31, . Difference 7 7 7 7 All the differences are the same so each term can be obtained by adding 7 to the previous term. The next term would be 31+= 7 38. (b) Here again the differences show a pattern. Sequence 3, 6, 11, 18, 27, . Difference 3 5 7 9 Here the differences increase by 2 each time, so to find further terms add 2 more than the previous difference. The next term would be 27+= 11 38. (c) The differences again help to see the pattern for this sequence. Sequence 1, 5, 6, 11, 17, 28, . Difference 4 1 5 6 11 If the first difference, 4, is ignored, the remaining pattern of the differences is the same as the sequence itself. This shows that the difference between any two terms is equal to the previous term. So a new term is obtained by adding together the two previous terms. The next term of the sequence will be 17+= 28 45. 265 12.2 MEP Pupil Text 12 Worked Example 3 A sequence of shapes is shown below. Write down a sequence for the number of line segments and explain how to find the next number in the sequence. Solution The sequence for the number of line segments is 4, 8, 12, 16, . The difference between each pair of terms is 4, so to the previous term add 4. Then the next term is 16+= 4 20. This corresponds to the shape opposite, which has 20 line segments. Exercises 1. Find the next two terms of each sequence below, showing the calculations which have to be done to obtain them. (a) 5, 11, 17, 23, 29, . (b) 6, 10, 15, 21, 28, . (c) 22, 19, 16, 13, 10, . (d) 30, 22, 15, 9, 4, . (e) 50, 56, 63, 71, 80, . (f) 2, 2, 4, 8, 14, 22, . (g) 6, 11, 16, 21, 26, . (h) 0, 3, 8, 15, 24, . (i) 3, 6, 12, 24, 48, . (j) 1, 4, 10, 22, 46, . (k) 4, −1, −11, − 26 , − 46, . 2. A sequence of numbers is 1, 8, 27, 64, 125, . By considering the differences between the terms, find the next two terms. 3. Each sequence of shapes below is made up of lines which join two points. For each sequence: (i) write down the number of lines, as a sequence; (ii) explain how to obtain the next term of the sequence; (iii) draw the next shape and check your answer. (a) 266 MEP Pupil Text 12 (b) (c) (d) (e) 4. Write down a sequence for the number of dots in each pattern. Then explain how to get the next number. (a) (b) (c) (d) 267 12.2 MEP Pupil Text 12 (e) (f) 5. (a) Describe how the sequence 1, 4, 9, 16, 25, 36. is formed. (b) What is the relationship between the sequence in (a) and the sequences below? For each sequence explain how to calculate the terms. (i) 3, 6, 11, 18, 27, 38, . (ii) −1, 2, 7, 14, 23, 34, . (iii) 2, 8, 18, 32, 50, 72, . (iv) 4, 16, 36, 64, 100, 144, . 6. (a) Describe how the sequences below are related. (i) 1, 1, 2, 3, 5, 8, 13, 21, . (ii) 4, 4, 5, 6, 8, 11, 16, 24, . (iii) −1, −1, 0, 1, 3, 6, 11, 19, . (b) Describe how to find the next term of each sequence. 7. A computer program prints out the following numbers. 1 2 4 8 11 16 22 When one of these numbers is changed, the numbers will form a pattern. Circle the number which has to be changed and correct it. Give a reason why your numbers now form a pattern. (SEG) 8. Here are the first four numbers of a number pattern. 7, 14, 21, 28, ... (a) Write down the next two numbers in the pattern. (b) Describe, in words, the rule for finding the next number in the pattern. (LON) 268 MEP Pupil Text 12 9. (a) To generate a sequence of numbers, Paul multiplies the previous number in the sequence by 3, then subtracts 1. Here are the first four numbers of his sequence. 1, 2, 5, 14, . Find the next two numbers in the sequence. (b) Here are the first four numbers of the sequence of cube numbers. 1, 8, 27, 64, . Find the next two numbers in the sequence. (MEG) 10. Row 1 12 19 28 39 52 q Row 2 791113p The numbers in Row 2 of the above pattern are found by using pairs of numbers from Row 1.