13 SEQUENCES W LO 1.1 Section title S E R C I P Neptune was the first planet to be found by mathematical prediction. It was found by looking at the number patterns of the other planets in the Solar System, and its position was correctly predicted to within a degree. Two scientists were eventually jointly credited with the discovery, one British and one French, but it has since been shown that the Brits took a bit too much of the credit! Objectives Before you start In this chapter you will: continue number patterns using the four rules of number continue patterns using pictures and give the rule for continuing the pattern use number machines to produce a number pattern and write down the rule complete a table of values using a number machine and write down the rule use the first difference to find the nth term and use the nth term to find any number in a sequence identify whether or not a number is in a sequence. You need to know: how to spot how a number pattern continues that the numbers in number patterns can get bigger and get smaller about odd numbers about even numbers that multiples are members of a multiplication table, e.g. 3, 6, 9, 12 are multiples of 3. 241 Chapter 13 Sequences 13.1 Sequences Objectives Why do this? You can continue number patterns by adding or subtracting, or multiplying or dividing by a number. You can give the term to term rules for continuing number patterns. You can continue patterns using pictures and give the rule for continuing the pattern. You may use sequences when learning a dance routine or a note sequence when playing a musical instrument. Get Ready Even numbers form a pattern Odd numbers also form a pattern 2, 4, 6, 8, 10, 12, ... They go up in twos. 1, 3, 5, 7, 9, 11, ... These also go up in twos. 1. Write down all the even numbers up to 20. 2. Write down all the odd numbers up to 20. 3. Check that you have written all the numbers from 1 to 20. Key Points A sequence is a pattern of numbers or shapes that follows a rule. Number patterns can be continued by adding, subtracting, multiplying and dividing. Patterns with pictures can be continued by finding the rule for continuing the pattern. The numbers in a number pattern are called terms. The term to term rules for continuing number patterns can be given. (This means you can say how you find a term from the one before it.) Continuing patterns by adding Example 1 a 14  4  18 18  4  22 a Write down the next two numbers in this number pattern. 2 6 10 14 b What is the rule you use to find the next number in the number pattern? c Find the 10th number in this pattern. 2 6 �4 10 �4 14 �4 18 �4 b To get the next number you add 4 each time. c The 10th number in the pattern is 38. 2 6 10 14 18 22 26 30 34 38 Carry on the number pattern until you get to the 10th number in the pattern. 242 terms term to term rules sequence 13.1 Sequences Questions in this chapter are targeted at the grades indicated. Exercise 13A 1 Find the two missing numbers in these number patterns. For each pattern, write down the term to term rule. a 3, 6, 9, __ , __ , 18, 21 b 3, 7, 11, __ , __ , 23, 27 c 5, 10, 15, 20, __ , __ , 35, 40 d 2, 7, 12, 17, __ , __ , 32, 37 e 1, 4, 7, 10, __ , __ , 19, 22 f 5, 7, 9, 11, __ , __ , 17, 19 g 3, 8, 13, 18, __ , __ , 33, 38 h 4, 7, 10, 13, __ , __ , 22, 25 i 2, 6, 10, 14, __ , __ , 26, 30 j 10, 20, 30, __ , __ , 60, 70 2 a Write down the next two numbers in these sequences. Examiner’s Tip i 1, 5, 9, 13, 17, … ii 2, 5, 8, 11, 14, … iii 3, 7, 11, 15, 19, … iv 4, 8, 12, 16, 20, … … means that the sequence v 5, 8, 11, 14, 17, … vi 5, 11, 17, 23, … carries on. vii 2, 6, 10, 14, 18, … viii 1, 7, 13, 19, 25, … ix 3, 11, 19, 27, 35, … x 5, 9, 13, 17, 21, … b Write down the rule you used to find the missing numbers in each sequence. 3 Find the 10th number of each of the number patterns in questions 1 and 2. 4 Jenny saves £2 each week in her piggy bank. Here is the pattern of how her money grows. Week 1 2 3 Money in piggy bank 2 4 6 F E 4 5 a Copy and complete the table. b Jenny is saving for a present for her Mum’s birthday that costs £20. How many weeks will this take? AO3 Continuing patterns by subtracting Key Point Number patterns can be continued by subtracting the same number from each term. Example 2 a Write down the next two numbers in this number pattern. 60 54 48 42 36 b What is the rule you use to find the next number in the number pattern? c Find the 8th number in this pattern. a 36  6  30 30  6  24 60 54 �6 48 �6 42 �6 36 �6 To get to the next number you take away 6. Take away 6 from 36 to get 30 then take away 6 from 30 to get 24. 243 Chapter 13 Sequences b To get the next number you subtract 6 each time. c The 8th number in the pattern is 18. 60 54 48 42 36 30 24 18 Carry on the number pattern until you get to the 8th number in the pattern. Exercise 13B F 1 Find the two missing numbers in these number patterns. Write down the rule for each number pattern. a 20, 18, 16, 14, __ , __ , 8 b 17, 15, 13, 11, __ , __ , 5 c 55, 50, 45, 40, __ , __ , 25 d 42, 37, 32, 27, __ , __ , 12 e 22, 19, 16, 13, __ , __ , 4 f 19, 17, 15, 13, __ , __ , 7 g 45, 38, 31, 24, __ , __ , 3 h 25, 22, 19, 16, __ , __ , 7 i 29, 25, 21, 17, __ , __ , 5 j 80, 70, 60, __ , __ , 30 2 a Write down the next two numbers in these sequences. i 41, 37, 33, 29, … ii 27, 24, 21, 18, … iii 59, 55, 51, 47, … iv 34, 31, 28, 25, … v 30, 27, 24, 21, … vi 61, 55, 49, 43, … vii 22, 20, 18, 16, … viii 51, 46, 41, 36, … ix 64, 57, 50, 43, … x 8, 6, 4, 2, 0, 2, … b Write down the rule you used to find the missing numbers in each sequence. E 3 Find the 10th number of each of the number patterns in questions 1 and 2. 4 Abdul’s mother gives him £20 each week to buy his lunch. His lunch costs him £3 each day. Here is the pattern of how he spends his money. Day M Tu Money left at end of day 17 14 W Th F a Copy and complete the table. AO3 244 b How much money will Abdul have left at the end of the week? 13.1 Sequences Continuing number patterns by multiplying Example 3 a Write down the next two numbers in this number pattern. 1 3 9 27 81 b What is the rule you use to find the next number in the number pattern? c Find the 8th number in this pattern. a 81  3  243 1 243  3  729 3 �3 9 �3 27 �3 81 �3 To get to the next number you multiply by 3. Multiply 81 by 3 to get 243 then multiply 243 by 3 to get 729. b To get the next number you multiply by 3 each time. c The 8th number in the pattern is 2187. 1 3 9 27 81 243 729 2187 Carry on the number pattern until you get to the 8th number in the pattern. Exercise 13C 1 Find the missing numbers in these number patterns. For each pattern, write down the rule. a 1, 2, 4, 8, __ , __ , 64 b 1, 4, 16, 64, __ , 1024 c 1, 5, __ , 125, __ , 3125 d 1, 10, 100, __ , __ , 100 000 e 3, 6, 12, 24, __ , __ , 192 f 2, 6, 18, __ , __ , 486 g 2, 8, 32, __ , __ , 2048 h 2, 20, 200, 2000, __ , __ , 2 000 000 i 2, 10, 50, __ , __ , 6250 j 3, 15, 75, __ , 1875 2 a Write down the next two numbers in these sequences. i 2, 4, 8, 16, … ii 3, 9, 27, 81, … iii 4, 16, 64, 256, … iv 5, 25, 125, 625, … v 5, 10, 20, 40, … vi 4, 12, 36, 108, … vii 10, 30, 90, 270, … viii 5, 50, 500, 5000, … ix 10, 20, 40, 80, … x 6, 36, 216, 1296, … b Write down the rule you used to find the missing number in each sequence. 3 Find the 10th number of each of the number patterns in questions 1 and 2. 4 The number of rabbits in a particular colony doubled every month for 10 months. The table shows the beginning of the pattern. Month 1 2 3 Number of rabbits 2 4 8 4 F E 5 a Copy and complete the table. b How many rabbits were in the colony in month 10? AO2 245 Chapter 13 Sequences Continuing number patterns by dividing Example 4 a Write down the next two numbers in this number pattern. 729 243 81 27 b What is the rule you use to find the next number in the number pattern? c Find the 8th number in this pattern. a 27  3  9 933 81 27 �3 9 �3 3 �3 1 �3 To get to the next number you divide by 3. Divide 27 by 3 to get 9 then divide 9 by 3 to get 3. b To get the next number you divide by 3 each time. 1 729 243 81 27 9 3 1 __ 3 Carry on the number pattern until you get to the 8th number in the pattern. 1 1  3  __ 3 1 c The 8th number in the pattern is 1  3  __ 3 Exercise 13D F E 1 Find the missing numbers in these number patterns. Write down the rule for each number pattern. a 64, 32, 16, 8, __ , __ , 1 b 1024, 256, 64, __ , 4 c 3125, 625, 125, __ , __ , 1 d 100 000, 10 000, 1000, __ , __ , 1 e 192, 96, 48, 24, __ , __ , 3 f 486, 162, 54, 18, __ , 2 g 1024, 512, 256, 128, __ , __ , 16 h 300 000, 30 000, 3000, __ , __ , 3 i 6250, 1250, 250, __ , __ , 2 j 2000, 200, 20, __ , __ , 0.02 2 a Write down the next two numbers in these sequences. i 64, 32, 16, 8, … ii 243, 81, 27, 9, … iii 128, 64, 32, 16, … iv 625, 125, 25, 5, … v 80, 40, 20, 10, … vi 972, 324, 108, 36, … vii 2430, 810, 270, 90, … viii 50 000, 5000, 500, 50, … ix 160, 80, 40, 20, … x 1296, 216, 36, 6, … b Write down the rule you used to find the missing number in each sequence. 3 Find the 8th number of each of the number patterns in questions 1 and 2. 4 The number of radioactive atoms in a radioactive isotope halves every 10 years. The table shows the beginning of the pattern. Years Number of atoms AO3 246 0 10 20 2560 1280 640 30 40 a Copy and complete the table. b How many radioactive atoms were in the isotope in year 100? 13.1 Sequences Continuing patterns in pictures Example 5 a Copy and complete the table for the number of matches used to make each member of the pattern. Pattern number 1 2 3 Number of matches used 4 7 10 4 5 6 7 b Write down the rule to get the next number in the pattern. c How many matches are there in pattern number 10? a Pattern number Number of matches used 1 2 4 7 3 4 5 6 7 10 13 16 19 22 Count the number of matches in each pattern and write down the number of matches used. 4 7 �3 10 �3 10  3  13 16  3  19 �3 �3 13  3  16 19  3  22 b Add 3 to the previous number. c Pattern number 10 has 31 matches. Continue the patterns. 22 25 28 31 Exercise 13E 1 For these patterns: i draw the next two patterns ii write down the rule in words to find the next pattern iii use your rule to find the 10th term. E a b c d e 247 Chapter 13 Sequences E 2 a Write down the number of matches in each of these patterns. Pattern 1 Pattern 2 Pattern 3 b Draw the next two patterns. c Write down the rule in words to continue the pattern. d Use your rule to find the number of matches needed for pattern number 10. 3 Repeat question 2 with the hexagon shape shown below. 13.2 Using input and output machines to investigate number patterns Objectives Why do this? You can use number machines to produce a number pattern and write down the rule (term number to term). You can complete a table of values using a number machine and write down the rule (term number to term). You can find missing values in a table of values and use the term number to term rule. When baking, you take your ingredients, mix them together and bake them in the oven, and you end up with a cake. Input Action This process applies to anything from baking a cake to making a motor car. Get Ready 1. Put the following numbers into this number machine and write down the answers. a 5 b 3 c 11 d 17 2. Draw a number machine for the process 6 and use it to find the answer when the following numbers are put into it. a 10 b 6 c 2 248 Output �3 13.2 Using input and output machines to investigate number patterns Key Points In this number pattern 3 Term 1 7 Term 2 11 Term 3 15 Term 4 19 23 … 3, 7, 11, 15, 19, 23, … are the terms. The term number tells you the position of each term in the pattern. In the sequence 3, 7, 11, 15, 19, … term 1 is 3, term 2 is 7, etc. You can use number machines to produce a number pattern and write down the rule (term number to term rule). You can complete a table of values using a number machine and write down the rule (term number to term). You can find missing values in a table of values and use the term number to term rule. Sometimes you can put two number machines together to make a sequence. One-stage input and output machines Example 6 This number machine has been used to produce the terms of a pattern. �5 a Complete the term numbers and terms in this table of values for the number machine. Term number Term 1 5 2 3 4 b What is the rule for working out the term from the term number? c Write down the rule for finding the next term from the term before it. a Term number 1 Term 5 Term number 1 Term 5 2 10 2 10 3 15 3 15 4 20 4 20 5 5 5 The rule for the number machine is multiply the term number by 5 so the terms will be 5, 10, 15, 20. b Multiply the term number by 5. c Add 5. To get to the term from the term number you multiply by 5. To get to the next term from the term before it you have to add 5 since the pattern is 5, 10, 15, 20, … table of values one-stage input and output machines 249 Chapter 13 Sequences Exercise 13F For each of these questions: a copy and complete the table of values for the number machine b write down the rule for finding the term from the term number c write down the rule for finding the next term from the term before it. F �3 Term number 1 2 3 4 Term 3 6 �7 Term number 1 2 3 4 Term 7 14 �4 Term number 1 2 3 4 Term 4 8 �2 Term number 1 2 3 4 Term 2 4 �8 Term number 1 2 3 4 Term 8 16 �10 Term number 1 2 3 4 Term 10 20 �12 Term number 1 2 3 4 Term 12 24 �50 Term number 1 2 3 4 Term 50 100 1 2 3 4 5 6 7 8 250 13.2 Using input and output machines to investigate number patterns Two-stage input and output machines Example 7 Term number �3 Term �1 This number machine has been used to produce the terms of a pattern. Examiner’s Tip Use the rule from the number machine on the term number to get to the term. You feed the result of the first machine into the second machine. a Complete the terms in a table of values for the number machine. Term number Term 1 4 2 3 4 b What is the rule for working out the term from the term number? c Write down the rule for finding the next term from the term before it. a Term number Term Term number Term Working 1 4 1 4 1314 2 7 2 7 2317 3 10 3 10 3  3  1  10 4 13 4 13 4  3  1  13 b Multiply by 3 and add 1. c Add 3. 3 3 3 To get to the term from the term number you 3 and 1. To get to the next term from the term before it you have to add 3 since the pattern is 4, 7, 10, 13, … 251 Chapter 13 Sequences Exercise 13G For each of these questions: a copy and complete the table of values for the number machine b write down the rule for finding the term from the term number c write down the rule for finding the next term from the term before it. E 1 �3 �2 Term number 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 Term 2 �2 �1 Term number Term 3 �3 �1 Term number Term 4 �4 �3 Term number Term 5 �5 �2 Term number Term 6 �3 �4 Term number Term 7 �3 �5 Term number Term 8 �5 �1 Term number Term 9 �4 �3 Term number Term 10 �5 �4 Term number Term 252 13.2 Using input and output machines to investigate number patterns Example 8 4 Complete this table of values for the number pattern with term number to term rule, ‘Multiply by 4 and subtract 2’. 2 2 Term number 1 6 2 6 2426 3 3 10 3  4  2  10 4 4 14 4  4  2  14 5 5 18 5  4  2  18 ↓ 30 8  4  2  30 ↓ 38 10  4  2  38 Term number 1 Term 2 ↓ Term Working 2 1422 ↓ ↓ 8 8 ↓ ↓ 38 10 ↓ You can find these terms by using the rule 4 then 2. Examiner’s Tip Don’t forget Bidmas: you do the  before the . You met Bidmas in Chapter 8. Exercise 13H Copy and complete these tables of values. 1 3 Term number 1 2 3 4 5 ↓ 10 ↓ 1 Term 4 ↓ ↓ 34 2 2 Term number 1 2 3 4 5 ↓ 10 ↓ 1 Term 1 ↓ ↓ 25 3 5 Term number 1 2 3 4 5 ↓ 10 ↓ 3 E Term 8 ↓ ↓ 78 253 Chapter 13 Sequences E 4 4 Term number 1 2 3 4 5 ↓ 10 ↓ 3 Term 1 ↓ ↓ 45 5 10 Term number 1 2 3 4 5 ↓ 10 ↓ 1 Term 11 ↓ ↓ 151 7 a Find the 10th number in this number pattern. 3, 7, 11, 15, … b What is the term number for the term that is 47? 8 a Find the 10th number in this number pattern. 4, 9, 14, 19, … b What is the term number for the term that is 69? 9 a Find the 10th number in this number pattern. 8, 11, 14, 17, … b What is the term number for the term that is 50? 6 5 Term number 1 2 3 4 5 ↓ 10 ↓ 3 Term 2 ↓ ↓ 67 13.3 Finding the nth term of a number pattern Objective Why do this? You can use the first difference to find the nth term of a number pattern and use the nth term to find any number in a number pattern or sequence. This may be useful when your teacher is dividing the class into groups, so that you can work out which group you are going to be in, or make sure you will be in a group with your friends. Get Ready 1. Write down the difference between each term in these number patterns. a 5, 10, 15, 20, 25, 30, … b 40, 35, 30, 25, 20, … c 4, 7, 10, 13, 16, … d 7, 11, 15, 19, 21, … e 50, 47, 44, 41, 37, … 2. Find the 10th term in each of the number patterns in question 1. Key Point The first difference can be used to find the nth term of a number pattern and then the nth term can be used to find any number in a sequence. 254 first difference 13.3 Finding the nth term of a number pattern Example 9 a Term number 1 Here is a number pattern 4, 7, 10, 13, 16, … a Find the nth term in this pattern. b Find the 20th term in this number pattern. Term Difference 4 2 3 4 5 7 10 13 16 n 3n  1 b The 20th term is 61. 3 3 3 3 Step 1 Put the number pattern into a table of values. Step 2 Find the difference between the terms in the number pattern. In this case it is 3. Step 3 Multiply each term number by the difference to get a new pattern. 3, 6, 9, 12, 15 … Step 4 Compare your new pattern with the original one and see what number you need to add or subtract to/from each term to get the original number pattern. In this case it is 1. The nth term is 3n  1. You replace the n by 20 in the nth term to find the 20th term. It is 3  20  1  61 Exercise 13I 1 For questions 1, 2 and 3 in Exercise 10H, find the nth term of each of the number patterns. 2 Write each pattern in a table and use the table to find the nth term of these number patterns. Use your nth term to find the 20th term in each of these number patterns. a 1, 3, 5, 7, 9, 11, … b 3, 5, 7, 9, 11, 13, … c 2, 5, 8, 11, 14, 17, … d 5, 8, 11, 14, 17, 20, … e 1, 5, 9, 13, 17, 21, … f 2, 6, 10, 14, 18, 22, … g 2, 7, 12, 17, 22, 27, … h 4, 9, 14, 19, 24, 29, … i 8, 13, 18, 23, 28, … j 5, 7, 9, 11, 13, … k 40, 35, 30, 25, 20, … l 38, 36, 34, 32, 30, … m 35, 32, 29, 26, 23, … n 20, 18, 16, 14, 12, … o 19, 17, 15, 13, 11, … p 190, 180, 160, 150, … C Examiner’s Tip To find the nth term of a sequence that gets smaller you subtract a multiple of n from a fixed number. e.g. 15  2n is the nth term of 13, 11, 9, 7, … 255 Chapter 13 Sequences C 3 Here is a pattern made from sticks. Pattern number 1 Pattern number 2 Pattern number 3 a Draw pattern number 4. b Copy and complete this table of values for the number of sticks used to make the patterns. Pattern number 1 2 Number of sticks 6 10 3 4 5 6 c Write, in terms of n, the number of sticks needed for pattern number n. d How many sticks would be needed for pattern number 20? 13.4 Deciding whether or not a number is in a number pattern Objective Why do this? You can use number patterns or use the nth term to identify whether a number is in the pattern. This is useful when you want to work out what will happen in the future, for example, you could work out whether next year will be a leap year as this happens every four years. Get Ready 1. Write each of these patterns in a table and use the table to find the nth term. Use your nth term to find the 20th term in each pattern. a 4, 7, 10, 13, … b 3, 8, 13, 18, … c 13, 15, 17, 19, … Key Points Number patterns or the nth term can be used to identify whether a number is in the pattern. Sometimes you will be asked how you know if a number is part of a sequence. You would then have to explain why the number is in the sequence or, even, why it is not in the sequence. 256 Chapter review Example 10 Here is a number pattern. 3 8 3 18 23 a Explain why 423 is in the pattern. b Explain why 325 is not in the pattern. a 423 is in the number pattern. Every odd term ends in 3 and goes up 3, 13, 23, etc, so 423 will be a member as it ends in a 3. b 325 is not in the number pattern. 325 ends in a 5 and every member of the pattern ends in either a 3 or an 8 so 325 cannot be in the pattern. There are other ways of answering questions like these. For example, you could identify the nth term The nth term is 5n  2 if 5n  2  423 5n  425 so n  85 so 423 is the 85th term. The nth term is 5n  2 so if 325 is in the pattern 5n  2  325 5n  327 so n  65.4 If 325 is in the pattern n must be a whole number. 65.4 is not a whole number so 325 is not in the pattern. Exercise 13J For each of these number patterns, explain whether each of the numbers in brackets are members of the number pattern or not. 1 1, 3, 5, 7, 9, 11, … (21, 34) 2 3, 5, 7, 9, 11, 13, … (63, 86) 3 2, 5, 8, 11, 14, 17, … (50, 66) 4 5, 8, 11, 14, 17, 20, … (50, 62) 5 1, 5, 9, 13, 17, 21, … (101, 150) 6 2, 6, 10, 14, 18, 22, … (101, 98) 7 2, 7, 12, 17, 22, 27, … (97, 120) 8 4, 9, 14, 19, 24, 29, … (168, 169) 9 40, 35, 30, 25, 20, … (85, 4) 10 38, 36, 34, 32, 30, … (71, 82) 11 3, 7, 11, 15, 19, 21, … (46, 79) 12 5, 11, 17, 23, 29, … (119, 72) AO3 C Chapter review A sequence is a number or shape pattern which follows a rule. Number patterns can be continued by adding, subtracting, multiplying and dividing. The term to term rules for continuing number patterns can be given. Patterns using pictures can be continued by finding the rule for continuing the pattern. You can use number machines to produce a number pattern and write down the rule (term number to term rule) You can complete a table of values using a number machine and write down the rule (term number to term). You can find missing values in a table of values and use the term number to term rule. The first difference can be used to find the nth term of a number pattern and then the nth term can be used to find any number in a sequence. Number patterns or the nth term can be used to identify whether a number is in the pattern. 257 Chapter 13 Sequences Review exercise F 1 Here are some patterns made of squares. Pattern number 1 Pattern number 2 Pattern number 3 The diagram below shows part of Pattern number 4. a Copy and complete Pattern number 4 Pattern number 4 AO2 D AO3 b Find the number of squares used for Pattern number 10 2 November 2008 adapted Here are the first 4 terms in a number sequence. 124 122 120 118 a Write down the next term in this number sequence. b Write down the 7th term in this number sequence. c Can 9 be a term in this number sequence? You must give a reason for your answer. AO3 AO3 May 2009 3 The nth term of a sequence is n2  4. Alex says ‘The nth term of the sequence is always a prime number when n is an odd number’. Is Alex correct? You must give a reason for your answer. November 2008 adapted 4 Here are the first 5 terms of a sequence. 1 1 2 3 5 The rule for the sequence is ‘The first two terms are 1 and 1. To get the next term add the two previous terms’. a Find the 6th term and the 7th term. b Find the 10th term. c Explain why after the first two terms the other terms of the sequence are alternately even and odd. The rule for another sequence is ‘The first two terms are 2 and 2. To get the next term multiply the two previous terms’. d Find an expression for the 10th term of this sequence. You do not have to wok out the expression. C AO3 5 Here are the first four terms of an arithmetic sequence. 5 8 11 14 Is 140 a term in the sequence? You must five a reason for your answer AO3 258 6 The first term of a sequence is x. To get the next term, multiply the previous term by 2 and add 1. The third term of the sequence is 21. Find the value of x.