nth term of a linear sequence Grade 4 nth term of a linear sequence Find the nth term of an arithmetic sequence and generate terms using it. If you have any questions regarding these resources or come across any errors, please contact helpful-report@pixl.org.uk

Lesson Plan Lesson Overview Progression of Learning Objective(s) Find the nth term of an arithmetic sequence and generate terms using it Grade 4 Prior Knowledge Substitution Basic operations Duration Allow 75 minutes to cover all aspects of this objective thoroughly Resources Print slides: 4, 9, 16, 20, 25 Equipment Progression of Learning What are the students learning? How are the students learning? (Activities & Differentiation) Substituting into a nth term to generate terms Give students slide 4 printed. Show how to substitute into 4n + 5 to generate the first 5 terms of the sequence. Repeat with 3n – 7. Students to complete 5 further practice questions. 10 What is a linear sequence Show students slide 8. Explain that a linear sequence has a first common difference. Students to then review the list of sequences on slide 4. Decide which sequences are linear and why. How to find the nth term of a linear sequence Give students slide 9 printed. Using slide 10, 11 and 12 demonstrate three different examples. Students to copy down all steps involved onto their sheet. Students to complete 5 further practice questions. 15 Using the nth term to find other terms in the sequence Give students slide 15 printed. Review a question – this is the standard layout in terms of possible questions. When asked to find the 11th term students can do this easily by counting along. How for part d when asked if 79 is in the sequence students need to rely on substituting into the nth term. Students to complete 5 further questions related to this aspect of the learning. Finding and using the nth term of linear sequences in problems involving angles. Give students slide 19 printed. Students to attempt questions independently initially. Then review collectively. Finding the nth term of an arithmetic sequence and generate terms using it in OCR exam questions (from specimen papers) Give students slide 24. This includes 2 exam questions related to objective. Students need to use notes from lesson to answer the questions. Ensure that all steps are shown. Relate to mark scheme to show how the marks are allocated. Next Steps Quadratic sequences Assessment PLC/Reformed Specification/Target 4/Algebra/Nth Term Of A Linear Sequence

Key Vocabulary Sequence Term nth term Linear sequence Arithmetic sequence Common difference

Generating terms from nth term Linear or Non Linear? (if linear what is the common difference?) 2, 4, 6, 8, 10 1, 4, 9, 16, 25, 36 1, 1, 2, 3, 5, 8 25, 21, 17, 13, 9 1, 100, 1001, 10000 6, 10, 14, 18, 22 2, 5, 10, 17, 26, 37 8n + 1 6n – 5 5n – 12 11 – 2n 4 – 3n 3n – 7 Student Sheet 1

Generating terms from nth term n represents the position number, so for the first 5 terms n = 1, 2, 3, 4 & 5 1) 4n + 5 When n = 1 4 x 1 + 5 = 9 When n = 2 4 x 2 + 5 = 13 When n = 3 4 x 3 + 5 = 17 When n = 4 4 x 4 + 5 = 21 When n = 5 4 x 5 + 5 = 25 So the first 5 terms are 9, 13, 17, 21, 25

Generating terms from nth term n represents the position number, so for the first 5 terms n = 1, 2, 3, 4 & 5 2) 3n – 7 When n = 1 3 x 1 – 7 = –4 When n = 2 3 x 2 – 7 = –1 When n = 3 3 x 3 – 7 = 2 When n = 4 3 x 4 – 7 = 5 When n = 5 3 x 5 – 7 = 8 So the first 5 terms are –4 , –1 , 2, 5, 8

Now you try… Write down the first 5 terms of the sequence whose nth term is given by: 9, 17, 25, 33, 41 1, 7, 13, 19, 25 –7, –2, 3, 8, 13 9, 7, 5, 3, 1 1, –2, –5, –8, –11

A linear or arithmetic sequence is where the terms increase or decrease by the same number. This is known as the common difference Look at these sequences 6, 13, 20, 27, 34 12, 10, 8, 6, 4 Common difference of +7 Common difference of –2 Both sequences are linear/arithmetic because they have the same common difference

Finding nth term of linear sequence 8, 11, 14, 17, 20 2, 10, 18, 26, 34 –8, –1, 6, 13, 20 4, 1, –2, –5, –8 –6, –8, –10, –12, –14 Find the 20th term 11, 15, 19, 23, 27 b) –1, 7, 15, 23, 31 Student Sheet 2

Finding nth term of linear sequence 6, 10, 14, 18, 22 5, 14, 23, 32, 41 15, 9, 3, –3, –9 8, 11, 14, 17, 20 2, 10, 18, 26, 34 –8, –1, 6, 13, 20 4, 1, –2, –5, –8 –6, –8, –10, –12, –14 Find the 20th term: 11, 15, 19, 23, 27 –1, 7, 15, 23, 31 Student Sheet 4

Finding nth term of linear sequence 1) 6, 10, 14, 18, 22 The sequence increases by 4, so the nth term starts with 4n +4 +4 +4 +4 Now compare the sequence to the 4 times table 6, 10, 14, 18, 22 Each term is 2 bigger than the 4 times table +2 +2 +2 +2 +2 So the nth term is 4n + 2 4, 8, 12, 16, 20

Finding nth term of linear sequence 2) 5, 14, 23, 32, 41 The sequence increases by 9, so the nth term starts with 9n +9 +9 +9 +9 Now compare the sequence to the 9 times table 5, 14, 23, 32, 41 Each term is 4 smaller than the 9 times table –4 –4 –4 –4 –4 So the nth term is 9n – 4 9, 18, 27, 36, 45

Finding nth term of linear sequence 3) 15, 9, 3, –3, –9 The sequence decreases by 6, so the nth term starts with –6n –6 –6 –6 –6 Now compare the sequence to the –6 times table 15, 9, 3, –3, –9 Each term is 21 bigger than the –6 times table +21 +21 +21 +21 +21 So the nth term is –6n + 21 –6, –12, –18, –24, –30

Finding nth term of linear sequence 8, 11, 14, 17, 20 2, 10, 18, 26, 34 –8, –1, 6, 13, 20 4, 1, –2, –5, –8 –6, –8, –10, –12, –14 3n + 5 8n – 6 7n – 15 –3n + 7 –2n – 4

Using nth term Find the 20th term 11, 15, 19, 23, 27 b) –1, 7, 15, 23, 31 nth term: 4n + 7 20th term, n = 20 4 x 20 + 7 = 87 nth term: 8n – 9 20th term, n = 20 8 x 20 – 9 = 151

Standard Question Format: Here are the first four terms of a number sequence. 7 11 15 Write down the next term in the sequence. Explain how you got your answer Work out the 11th term in the sequence Is 79 a term in this sequence? Explain how you got your answer 3 10 17 24 31 150? 9 15 21 27 5 8 11 14 34? 23 29 35 149? 2 6 18 87? Students have a go on their own. Focus area will be part c Student Sheet 3

Standard Question Format: Here are the first four terms of a number sequence. 3 7 11 15 Write down the next term in the sequence Explain how you got your answer (b) Work out the 11th term in the sequence (c) Is 79 a term in this sequence? 19 Add 4 to the term before Students have a go on their own. Focus area will be part c 23, 27, 31, 35, 39, 43

1. Look for an obvious pattern 2. Find nth term, solve is n integer? 3 7 11 15 Is 79 a term in this sequence? 1. Look for an obvious pattern 2. Find nth term, solve is n integer? 4n - 1 = 79 4n = 80 n = 20 Yes, 79 is the 20th term

7n - 4 3 10 17 24 31 150? 9 15 21 27 5 8 11 14 34? 23 29 35 149? 2 6 18 87? 6n - 3 3n + 2 6n + 5 4n - 2 No explaination just check and see how you did

Problem Solving & Reasoning Here are three patterns using circles Draw pattern 4 Write down an expression for the nth pattern How many circles are in the 50th pattern Kelly has £200 in her Post Office savings account. Each month she will add an extra £15 to the account. Kelly wants to save more than £480. How long will this take? Here are the first 5 terms of an arithmetic sequence: 3, 10, 17, 24, 31 Show that 220 is in the sequence. Student Sheet 4

Sequences from Patterns Here are three patterns using circles Draw pattern 4 Write down an expression for the nth pattern How many circles are in the 50th pattern Pattern 1 Pattern 2 Pattern 3

Solution 3, 5, 7, 9 nth term: 2n + 1 nth term: 2n + 1 and n = 50 Pattern 1 Pattern 2 Pattern 3 (a) Pattern 4 (b) The nth term The number of circles in each pattern forms a linear sequence 3, 5, 7, 9 nth term: 2n + 1 (c) Number of circles in the 50th term nth term: 2n + 1 and n = 50 101 circles in 50th pattern 2 x 50 + 1 = 101

Problem Solving and Reasoning Here are the first 5 terms of an arithmetic sequence: 3, 10, 17, 24, 31 Show that 220 is in the sequence. nth term: 7n – 4 Remember n represents the position number of the sequence. Therefore n must be an integer. Form the equation 7n – 4 = 220 7n = 224 n = 224 ÷ 7 n = 32 So 220 is the 32nd term in the sequence

Problem Solving and Reasoning Kelly has £200 in her Post Office savings account. Each month she will add an extra £15 to the account. Kelly wants to save more than £480. How long will this take? Here is the sequence: 215, 230, 245, 260, 275...... nth term is 15n + 200 We want more than £480 So we need to solve the following inequality 15n + 200 > 480 15n > 280 n > 280 ÷ 15 n > 18.6667 Because n is an integer it will take Kelly 19 months to save over £480.

Exam Questions – Specimen Papers Student Sheet 5

Exam Questions – Specimen Papers

Exam Questions – Specimen Papers