Find terms of a sequence and say whether it is ascending or descending, finite or infinite Find the next term in a sequence of numbers or shapes

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1 1.1 Sequences Find terms of a sequence and say whether it is ascending or descending, finite or infinite Find the next term in a sequence of numbers or shapes Key words sequence term consecutive infinite finite A number sequence is a set of numbers in a given order, such as 1, 4, 7, 10, Each number in a sequence is called a term. Terms next to each other are called consecutive terms. Sequences may be ascending (e.g. 2, 4, 6, 8, ) or descending (e.g. 18, 15, 12, 9, ). The sequence 1, 2, 3, 4, 5... is infinite. We could go on counting forever. The sequence 10, 12, is finite. The dots mean that there are missing terms and that the sequence continues in the same way until the final value, 98, is reached. Example 1 Here is a sequence of diagrams. a) Spot the pattern and draw the next two terms in the sequence. b) Is the sequence finite or infinite? a) Each time two more dots are added. This is the fifth term. This is the sixth term. b) The sequence is infinite. Example 2 The first term of a sequence is 12; each term in the sequence is found by subtracting 4 from the previous term. a) Write down the first five terms of the sequence. b) Is this sequence ascending or descending? First term 12 a) 12, 8, 4, 0, 4 Second term Third term b) The sequence is descending. Fourth term Fifth term Exercise Spot the pattern and draw the next two terms in each sequence. a) b) c) d) 1st term 2nd term 3rd term 2 Maths Connect 1R

2 Decide whether each of the sequences in Q1 are finite or infinite. Find the next two terms of each of the following sequences and decide whether they are ascending or descending: a) 6, 9, 12, 15, b) 0.5, 1, 1.5, 2, 2.5, 3, c) 5, 50, 500, 5000, d) 128, 64, 32, 16, e) 11, 9, 7 Write down the first five terms of each of the following sequences: a) The first term is 12 and each term is found by adding 4 to the previous term. b) The first term is 81 and each term is found by dividing the last term by 3. c) The second term is 12 and each term is found by subtracting 1 from the previous term. d) The fifth term is 17 and each term is found by adding 4 to the previous term. Copy and complete these sequences of numbers and describe the pattern you are using to move from term-to-term: a) 2,, 10,, 18,, 26 b) 15,,, 6,,, 3 c) 1,, 100,, d) 1 2,, 1 8 1,, 3 12,, 1 28 To make one beach-hut out of matchsticks we used six matchsticks. To make two beach-huts we used an extra five (eleven matchsticks in total). a) Copy and complete the table below: b) Predict how many matchsticks you would need for ten beach-huts. c) How many lots of five matchsticks did you need for ten beach-huts? d) How many matchsticks would you need for 70 beach-huts? e) Describe how you calculated the number of matchsticks needed for 70 beach-huts. The first few terms of a sequence are: 3, 6, 12, 24. How many terms are less than 200? Each term of a sequence is found by adding 5 to the previous term. a) i) If the first term is 5 what is the sixth term? ii) How many lots of 5 is this? b) i) If the first term is 6 what is the seventh term? ii) How many lots of 5 is this? Investigation Number of beach-huts Number of matchsticks 6 11 Look at the pattern for moving from the 1st to the 3rd term, the 3rd to the 5th and so on. How can you adapt this pattern for moving from term-to-term? How many lots of five matchsticks do you need? The first term of a sequence is 100. To find the next term you divide by 2. a) Write down the first few terms of the sequence. b) Replace divide by 2 with add 2, subtract 2 or multiply by 2 and find the first few terms of each sequence. c) When the pattern was: i) divide by 2 ii) multiply by 2 iii) add 2 iv) subtract 2 was the sequence ascending or descending? d) Will this also be true if you replace 100 and 2 with other numbers? Sequences 3

3 1.2 Generating sequences Key words generate term Generate a sequence given a starting point and a rule to go from term-to-term rule term-to-term Use the rule to find a term in a sequence without finding all the values in between. Generating a sequence means writing down the terms of the sequence. To do this you need to know the pattern that the sequence follows. To generate a sequence you may be given a starting point and a rule that connects one term to the next. This is called the term-to-term rule. For example, for this sequence: 5, 11, 17, 23, 29, 35, the first term of the sequence is 5 and the term-to-term rule is add 6. Since we add 6 each time, we can compare the sequence to the multiples of Sequence: Multiples of 6: We can see that this sequence is the multiples of 6 minus 1. We can write this sequence like this: 1st term nd term rd term th term th term By following the pattern it is easy to find any term in the sequence. For example, the 65th term is Example A sequence starts with 2 and the term-to-term rule is add 4. a) Find the first five terms of the sequence. First Term 2 b) Find the tenth term of the sequence without Second Term working out all the terms in between. Third Term Fourth Term a) 2, 6, 10, 14, 18 Fifth Term Exercise Write down the first five terms of a sequence given the following first terms and rules. First term Term-to-term rule a) 7 Add 4 b) 3 Add 2 c) 8 Add 3 d) 100 Subtract 6 e) 74 Subtract 12 4 Maths Connect 1R 5, 11, 17, 23, 29, 35, 6, 12, 18, 24, 30, 36, b) Each term is 2 less than the multiples of 4. So the tenth term will be Since we add 4 each time we can compare this sequence to the multiples of 4: Sequence 2, 6, 10, 14, 18,... Multiples of 4 4, 8, 12, 16, 20,...

4 Look at Q1. Fill in the gaps in the following sentences: The sequence in Q1a is the multiples of add. The sequence in Q1b is the multiples of add. The sequence in Q1c is the multiples of add. Write down the ninth term of the sequences in Q1a, b and c. The first term of a sequence is 30. The term-to-term rule is subtract 2. a) Write down the first five terms. b) How many more times will you have to subtract 2 to get from the fifth term to the seventh term? c) Write down the tenth term without continuing the sequence. Look at the sequence in Q1d. a) How many times will you have to subtract 6 to get from the fifth term to the seventh term? b) Write down the tenth term without continuing the sequence. The second term of a sequence is 5. The term-to-term rule is add 4. a) Find the first term. b) Find the 100th term. The term-to-term rule is multiply by 2. Copy and complete the sequence:,,,, 88 Write down the 15th and 21st terms of the following sequences: a) The first term is 36, to get to the next term add 12 b) The first term is 0, to get to the next term add 5 c) The first term is 8, to get to the next term add 0.5. Investigation The first two terms in a sequence are: 1, 2 a) Find and describe a term-to-term rule connecting these terms and then write the next three terms in the sequence. b) There is another rule that could connect the first two terms of this sequence, what is it? Write down the first five terms of the sequence using the second term-toterm rule. c) Randomly pick two numbers between 1 and 10 as the first two terms of a sequence. List as many term-to-term rules as you can for the sequence, and write the next three terms for each sequence. d) If you are only given the first two terms of a sequence can you always find more than one rule? e) If you are given the first three terms of a sequence, can there be more than one rule which connects the terms? If you think this is true explain why it is true, if you think this is false show an example where it doesn t work this is called a counter example Generating sequences 5

5 1.3 The general term Generate a sequence given a general term Find the general term of a sequence Key words term term number general term We can describe a sequence of numbers using the first term and a term-to-term rule. For example, the sequence whose first term is 4 and term-to-term rule is add 3 is 4, 7, 10, 13, Another way of describing a sequence is by giving a rule that connects the term number and the term. We call this the general term. Look at the sequence 8, 10, 12, 14, 16. The difference between terms is 2 so we compare it to the multiples of 2. The sequence is the multiples of 2 add 6. 1st term nd term rd term th term General term term number 2 6 You can use the general term to find the value of any term in the sequence. You substitute the term number into the general term. For example, the 20th term of this sequence is Example a) Find the general term of the sequence 5, 7, 9, 11, 13, b) Find the 16th term of the sequence. a) 1st term nd term th term We can see that this sequence is the multiples of 2 add 3. The general term is: term number 2 3 b) The 16th term is Look for a connection between the term number and the term. Substitute the term number into the general term. Exercise For each of these sequences: a) Find the general term. b) Find the 100th term. Term Number Sequence A Sequence B Sequence C Sequence D Sequence E Sequence F Maths Connect 1R

6 Write down a general term for each of the following sequences. Then work out the tenth term in each sequence. a) 3, 8, 13, 18, b) 99, 198, 297, 396, c) 76, 151, 226, 301, d) 0.1, 0.2, 0.3, 0.4, e) 40, 75, 110, 145, The first five patterns of some sequences are shown in the table. For each sequence: a) How many extra dots are added each time? b) Write down the general term for the pattern. Term Number Pattern A Pattern B Pattern C Pattern D Two rival taxi firms have the following rates: Rovers Rides 3.00 basic charge 50p per mile Candice Cars 2.50 basic charge 60p per mile. a) Find the cost of each firm for journeys of 1, 2, 3, 4 and 5 miles. b) Which company would you use for a journey of 1 mile? c) Which company would you use for a journey of 10 miles? d) Find the general term for each taxi firm s rates. The bill for a mobile phone is calculated using the table below: Duration of call (minutes) Cost ( ) a) Find the general term for the cost of the phone bill. b) Calculate the phone bill for a call lasting 20 minutes. Investigation Look at the differences between consecutive terms in each of the sequences in Q1, Q2 and Q3. a) What is the connection between the difference and the general term for the sequence? b) Write a set of instructions for finding the general term of a sequence. The general term 7

7 1.4 Function machines Find unknown numbers and operations in function machines Draw simple mapping diagrams Key words function machine input output inverse unknown mapping diagram A function machine changes one number (the input ) to another number (the output ) according to a rule. For example: Input 4 Output Here the rule is add 4. The operation 4 is performed on any input. You can use any of these operations in a function machine: addition, subtraction, multiplication or division. You can use the inverse operation to work backwards through the function machine to find an unknown input. We can use a mapping diagram to show how one number moves to another number using a rule. For the function machine above, the mapping diagram is: Example 1 Find the unknown number in these function machines: a) 3 27 b) 20 5 a) 27 3, so 9 Check: b) , so 4 Check: You need to work backwards. The inverse of multiplication is division. Always check your answer. Example 2 Input 7 2 Output a) What is the output if the input is 3? b) What is the input if the output is 37? Function machines can have more than one step. a) Output b) So Input 5 Work backwards and use inverse operations. 8 Maths Connect 1R

8 Exercise Draw mapping diagrams for parts a) to d) and find the rules that connect the pairs of numbers. a) 1 8 b) 3 12 c) 12 7 d) Draw function machines for the rules in Q1. Copy and complete the function machines below: a) 8 2 b) c) 5 7 d) 42 7 Draw function machines to represent the following statements. Find the original number in each case. a) Twice this number is 102. b) Three more than this number is 20. c) 12 less than this number is 79. d) Half this number is 35. Find the missing inputs in these two-step function machines: a) Input b) Input c) Input d) Input What number am I thinking of? a) I think of a number, divide by 4, add 3 and get 28. b) I think of a number, subtract 6, multiply by 5 and get 0. c) I think of a number, divide by 2, multiply by 3 and get 30. Draw function machines to help you. Tom s calculator isn t working properly. When he enters a number it multiplies it by 10 and subtracts 11. a) If he enters the number 30, what will the calculator show? b) If the calculator shows the number 9, what number did Tom enter? c) If Tom enters 7 8, what will the calculator show on its screen? d) If the calculator shows the number 1989, what number did Tom enter? Copy and complete the following function machines using the numbers written next to them. The first one is done for you. a) 2, 7, 9, b 1, 3, 5, 8 c) 2, 5, 6, 7 Are these two function machines identical? Explain your answer. Experiment with different inputs. Input 5 9 Output Input 9 5 Output Replace 5 and 9 with other operations (e.g. 2, 12, 6) and decide whether the order in which the operations are carried out affects the output. Function machines 9

9 1.5 More function machines Write simple expressions and equations using letters to stand in for unknown numbers Solve simple equations by converting them into function machines Key words algebra unknown expression equation In maths we often use a letter to stand in for a number we don t know. This is called algebra. If you chose a number and added 5, you could represent this by writing: x 5 It doesn t matter which letter you use. It is just a method of writing an unknown number. We call x 5 an algebraic expression. The sentence I choose a number, subtract 10 and get 12 can be written: x This is called an algebraic equation since it contains an unknown (x) and an sign. The same rules and conventions that you use in arithmetic also apply in algebra. Example 1 Write the following function machines as algebraic equations and find the unknowns: a) Input b) Input a) m m Check: b) 3x x 27 x We can choose any letter to represent an unknown. Use the inverse operation to find the unknown. 3x is the same as 3 x or x 3. Example 2 Write each statement as an algebraic equation and find the unknown: a) I think of a number, multiply it by 5, add 2 and get 27. b) I think of a number, divide it by 4 and get 12. a) 5b b 25 b 5 x b) 12 4 x Represent the unknown number with a letter, in this case b. In algebra we write x x 4 as 4 10 Maths Connect 1R

10 Exercise Georgina, Raashad and Emily are all given the same amount of money. Use the letter b to represent the number of pence they are given. a) Georgina spends 45p of hers. How much money does she have now? b) Raashad shares his with his brother. How much money does he have now? c) Emily is given 1 more. How much money does she have now? Write the following statements as algebraic expressions: a) I choose a number and subtract 5. b) I choose a number and triple it. c) I choose a number, multiply it by 5 and add 14. d) I choose a number divide it by 7 and subtract 10. Choose a letter to stand in for the unknown number. Write the following as algebraic equations and calculate the number I choose. a) I choose a number, add 12 and get 17. b) I choose a number, divide it by 5 and get 12. c) I choose a number, multiply it by 12 and get 48. d) I choose a number, subtract 0.5 and get 1.5. e) I choose a number, multiply it by 100, subtract 12 and get f) I choose a number, divide it by 8, subtract 10 and get 1. Write the following function machines as algebraic equations and find the missing inputs: a) Input b) Input c) Input d) Input A set of numbers is mapped to another set of numbers using the following function machine: Input Output a) Find the output if the input is 5. b) Find the input if the output is 5. c) Write an algebraic expression connecting the input to the output. d) Work out the outputs for inputs of 1, 2, 3, 4 and 5. Explain any pattern you see. Use one letter for the input and another for the output. Look at the difference between consecutive terms in the sequence of answers. Ian s pocket money (in pence) is calculated by doubling his age and adding 50p. Gaby s pocket money (in pence) is calculated by adding her age to 70p. a) Calculate the amounts of pocket money Ian and Gaby receive when they are 12. b) Calculate Ian s age when he receives 78p pocket money. c) Calculate Gaby s age when she receives 78p pocket money. Investigation In Q6, Ian was born in 1990 and Gaby was born in Investigate the amount of pocket money they will receive each year until they are 18. In which year will they both receive exactly the same amount of pocket money? What will their ages be then? More function machines 11

11 1.6 Using letter symbols to stand in for unknown numbers Use letters to stand in for unknown numbers Express relationships between unknowns Key words unknown relationship algebra You already know that you can use a letter to stand in for an equation, for example 5x unknown number in an Sometimes we want to express one unknown number in terms of its with another unknown. For example, if Mary gets twice as much money as Nuala, this can be expressed using algebra as M 2N We can use algebra to express a function machine: relationship M is the amount of money Mary receives, N is the amount of money Nuala receives. Input 8 5 Output Using algebra, we write: x 5 y 8 x represents the input and y represents the output. Example 1 Express the relationships between the following pairs of letters using algebra. a) A is twice B. b) D is four less than N. c) M is five more than triple T. d) Q is three times smaller than D. a) A 2B b) D N 4 c) M 3T 5 d) Q D 3 We show things have the same value by using the sign. Example 2 If Katy received 10p more pocket money and Asif received 50p less they would receive the same amount. Use algebra to show the relationship between the amounts of pocket money Katy and Asif receive. p 10 q 50 Choose a letter symbol to stand in for the amounts of pocket money Katy and Asif receive. Here p is the amount Katy receives and q is the amount Asif receives. 12 Maths Connect 1R

12 Exercise Use algebra to show the relationships between the following pairs of letters: a) M is three less than T. b) X is four more than Z. c) Y is seven less than twice V. d) Q is four times smaller than R. e) X is four more than five times W. f) Z is seven times smaller than B. Use algebra to show the relationship in each of the following situations: a) James is four years younger than Andrea. b) Georgina receives half the amount of pocket money Raashad receives. c) If Angharad spent 10 minutes more on the phone she would spend half the amount of time that Emily does. Write the following algebraic expressions in another form: We can write: 3 5 as a) 2x b) x x x c) b b b b b d) 7m e) 3z If possible, write the following function machines using only one box. If it is not possible, say why not. When you have done this, express each function machine using algebra and find the input for each one. a) Input b) Input c) Input d) Input e) Input f ) Input Two whole numbers add together to make 15. The two numbers multiplied together give 50. a) Use algebra to express this information. b) Find the numbers. Try all the pairs of numbers that multiply together to give 50. In this Arithmagon the number in each square is the sum of the numbers in the two circles on either side of it. a) Find as many ways as you can to express relationships between the letters using algebra. A 17 B b) Find the values of B, C and D if A 10. c) Investigate other values for A, B, C and D. You should find many sets of values that work C 21 D Investigation In this diagram the value in each cell is found by adding the values in the two cells above it. a) Copy and complete the diagram. a b c b) Does the value in the bottom cell change if you change the a b order of a, b and c in the top row? c) Draw a similar diagram with four unknowns a, b, c and d. Investigate the value in the bottom cell for different arrangements of a, b, c and d in the top row. Using letter symbols to stand in for unknown numbers 13