SEQUENCES AND SERIES Unit 2 General Maths DESCRIBING SEQUENCES This topic investigates different types of patterns and how they can be manipulated mathematically. The dictionary describes a sequence as, ‘a number of things, actions, or events arranged or happening in a specific order or having a specific connection’. In maths, the term sequence is used to represent an ordered set of elements. In this topic we will examine the relationships and patterns of these sets of data. RECOGNISING PATTERNS For these examples: eg1) 2, 4, 6, 8…….. Describe the pattern in words; Describe the pattern in mathematical terms; State the next 3 numbers in the pattern. Increasing by 2 +2 10, 12, 14 Doubling each number x2 80, 160, 320 eg2) 5, 10, 20, 40….. eg3) 1000, 500, 250…… Halving each number ÷2 125, 62.5, 31.25 USING A RULE TO GENERATE A NUMBER PATTERN For these examples: eg1) Use the following rules to write down the first five numbers of each number pattern Start with a 72 and divide by 2 each time. 72, 36, 18, 9, 4.5 eg2) Start with 2, Multiply by 4 and subtract 3. 2, 5, 17, 65, 257 NOW DO Chapter 5 Exercise 5A Questions 1 – 3 ARITHMETIC SEQUENCES An ARITHMETIC SEQUENCE is one in which the difference between any two consecutive terms is the same. Are these Arithmetic Sequences? eg. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 Yes – the difference between consecutive terms is the same eg. 0, 3, 6, 9, 12, 15, 18, 21, 24 Yes – the difference between consecutive terms is the same eg. 2, 4, 5, 10, 11, 16 No – the difference between consecutive terms is NOT the same eg. 2, 4, 8, 16, 32, 64 No – the difference between consecutive terms is NOT the same ARITHMETIC SEQUENCES The terms, in order, can be labelled 𝑡1 , 𝑡2 , 𝑡3 , … … 𝑡𝑛 Label the terms in the following examples: eg. 2, 3, 4, 5, 6, 7, 8, 9, 10 𝑡1 = 2, 𝑡2 = 3, 𝑡3 = 4 𝑒𝑡𝑐 … eg. 0, 3, 6, 9, 12, 15, 18, 21, 24 𝑡1 = 0, 𝑡2 = 3, 𝑡3 = 6 𝑒𝑡𝑐 … eg. 𝑡1 = 4, 𝑡2 = 6, 𝑡3 = 8 𝑒𝑡𝑐 … 4, 6, 8, 10, 12, 14 ARITHMETIC SEQUENCES We can label the first term in the sequence ′𝑎′ We can label the ‘common difference’ between consecutive terms Label the arithmetic sequences with their ‘a’ and ‘d’ values 𝑎=2 eg. 2, 3, 4, 5, 6, 7, 8, 9, 10 eg. 0, 3, 6, 9, 12, 15, 18, 21, 24 eg. 4, 6, 8, 10, 12, 14 𝑑=1 𝑎=0 𝑎=4 𝑑=3 𝑑=2 ′𝑑′ ARITHMETIC SEQUENCES GENERATED BY RECURSION A sequence can be generated by the repeated use of an instruction. This is known as recursion. We can form equations to model recursion: The current term given by 𝑛 can be represented by 𝑡𝑛 The following term is 𝑡𝑛+1 & the term before it is 𝑡𝑛−1 Using this, we can form an equation to model the recursion. The recursion relation is: 𝑡𝑛+1 = 𝑡𝑛 + 𝑑 , The next term The current term 𝑡1 = 𝑎 The common difference The first term in the sequence Recursion relation The next term 𝑡𝑛+1 = 𝑡𝑛 + 𝑑, The current term 𝑡1 = 𝑎 The common difference Form recursion relations for the following: eg. 2, 3, 4, 5, 6, 7, 8, 9, 10 𝑡𝑛+1 = 𝑡𝑛 + 1, 𝑡1 = 2 eg. 0, 3, 6, 9, 12, 15, 18, 21, 24 𝑡𝑛+1 = 𝑡𝑛 + 3, 𝑡1 = 0 eg. 𝑡𝑛+1 = 𝑡𝑛 + 0.5, 𝑡1 = 4 𝑡𝑛+1 = 𝑡𝑛 − 3, 𝑡1 = 30 4, 4.5, 5, 5.5, 6, 6.5 eg. 30, 27, 24, 21, 18, 15 The first term in the sequence NOW DO Chapter 5 Recursion Worksheet – Exercise 1 FINDING TERMS OF AN ARITHMETIC SEQUENCE If we know the first term 𝑎, and the common difference 𝑑, we can find any number within the arithmetic sequence. The rule for finding a term in an arithmetic sequence is: 𝑡𝑛 = 𝑎 + 𝑛 − 1 𝑑 The value of the nth term (the term we are trying to find) The first term in the sequence The common difference The position of the term we are trying to find in the sequence Rule for Arithmetic Sequences The value of the n th term (the term we are trying to find) eg. Consider the arithmetic sequence 𝑡𝑛 = 𝑎 + 𝑛 − 1 𝑑 The first term in the sequence The common difference The position of the term we are trying to find in the sequence 22, 28, 34, 40, ….. a) Write a rule for the arithmetic sequence 𝑡𝑛 = 𝑎 + 𝑛 − 1 𝑑 𝑡𝑛 = 22 + 𝑛 − 1 6 b) What number would be at the 30th position? 𝑡𝑛 = 22 + 𝑛 − 1 6 𝑡30 = 22 + 30 − 1 6 𝑡30 = 22 + 29 6 = 22 + 174 = 196 Rule for Arithmetic Sequences The value of the n th term (the term we are trying to find) eg. Consider the arithmetic sequence 𝑡𝑛 = 𝑎 + 𝑛 − 1 𝑑 The first term in the sequence The common difference The position of the term we are trying to find in the sequence 10, 13, 16, 19, 22, 25,… a) Write a rule for the arithmetic sequence 𝑡𝑛 = 𝑎 + 𝑛 − 1 𝑑 𝑡𝑛 = 10 + 𝑛 − 1 × 3 b) What number would be at the 16th position? 𝑡𝑛 = 10 + 𝑛 − 1 × 3 𝑡16 = 10 + 16 − 1 × 3 𝑡16 = 10 + 15 × 3 = 10 + 45 = 55 Rule for Arithmetic Sequences The value of the n th term (the term we are trying to find) eg. Consider the arithmetic sequence 𝑡𝑛 = 𝑎 + 𝑛 − 1 𝑑 The first term in the sequence The common difference The position of the term we are trying to find in the sequence 41, 37, 33, 29, 25,…. a) Write a rule for the arithmetic sequence 𝑡𝑛 = 𝑎 + 𝑛 − 1 𝑑 𝑡𝑛 = 41 + 𝑛 − 1 × −4 b) What number would be at the 11th position? 𝑡𝑛 = 41 + 𝑛 − 1 × −4 𝑡30 = 41 + 11 − 1 × −4 𝑡30 = 41 + 10 × −4 = 41 − 40 = 1 NOW DO Chapter 5 Exercise 5B Questions 1, 4a, 4b, 4d 4f, 5a, 5b, 5d, 5f ARITHMETIC SERIES An arithmetic series is the term used for the sum of all of the terms in an arithmetic sequence. We can find the arithmetic series using one of two different methods. If we know the first term 𝑎, and the last term 𝑙, the sum of the 𝑛 terms is given by: The number of terms in the sequence 𝑛 𝑆𝑛 = ( 𝑎 + 𝑙 ) 2 The first term in the sequence The last term in the sequence ARITHMETIC SERIES An arithmetic series is the term used for the sum of all of the terms in an arithmetic sequence. We can find the arithmetic series using one of two different methods. If we know the first term 𝑎, the common difference 𝑑, the sum of the 𝑛 terms is given by: The number of terms in the sequence 𝑛 𝑆𝑛 = ( 2𝑎 + 𝑛 − 1 𝑑) 2 The first term in the sequence The common difference The number of terms in the sequence eg. A sequence starts at 4 and ends with 32. If there are 8 terms in the sequence, find the arithmetic series (the sum) for the sequence. 𝑆𝑛 = 𝑛 2 = 8 2 𝑛 𝑆𝑛 = ( 2𝑎 + 𝑛 − 1 𝑑 2 The first term in the sequence The first term in the sequence (𝑎+𝑙) 4 + 32 = 4( 36) = 144 The common difference The number of terms in the sequence 𝑛 𝑆𝑛 = ( 𝑎 + 𝑙 ) 2 The last term in the sequence The number of terms in the sequence eg. Find the sum of the first 20 values in the sequence 3, 5, 7, 9,… 𝑆𝑛 = = 20 2 𝑛 2 𝑛 𝑆𝑛 = ( 2𝑎 + 𝑛 − 1 𝑑 2 The first term in the sequence The common difference ( 2𝑎 + 𝑛 − 1 𝑑 The first term in the sequence 2 × 3 + 20 − 1 2 = 10(6 + 19 × 2 ) = 10 6 + 38 = 10 × 44 = 440 The number of terms in the sequence 𝑛 𝑆𝑛 = ( 𝑎 + 𝑙 ) 2 The last term in the sequence The number of terms in the sequence eg. Find the sum of the first 18 values in the sequence 6, 11, 16, 21… 𝑆𝑛 = = 18 2 𝑛 2 𝑛 𝑆𝑛 = ( 2𝑎 + 𝑛 − 1 𝑑 2 The first term in the sequence The common difference ( 2𝑎 + 𝑛 − 1 𝑑 The first term in the sequence 2 × 6 + 18 − 1 5 = 9(12 + 17 × 5 ) = 9 12 + 85 = 9 × 97 = 873 The number of terms in the sequence 𝑛 𝑆𝑛 = ( 𝑎 + 𝑙 ) 2 The last term in the sequence The number of terms in the sequence eg. Find the sum of the first 10 values in the sequence 40, 36, 32, 28,… 𝑆𝑛 = = 10 2 𝑛 2 𝑛 𝑆𝑛 = ( 2𝑎 + 𝑛 − 1 𝑑 2 The first term in the sequence The common difference ( 2𝑎 + 𝑛 − 1 𝑑 The first term in the sequence 2 × 40 + 10 − 1 × −4 = 5(80 + 9 × −4 ) = 5 80 − 36 = 5 × 44 = 220 The number of terms in the sequence 𝑛 𝑆𝑛 = ( 𝑎 + 𝑙 ) 2 The last term in the sequence NOW DO Chapter 5 Exercise 5C Questions 1, 4, 6, 7, 8, 9a ARITHMETIC SERIES We can also use the arithmetic series to find the value of a term at any position 𝑛 within the sequence. Value of the term n 𝑡𝑛 = 𝑆𝑛 − 𝑆𝑛−1 Arithmetic Series until the term n Arithmetic series until term before n Value of the term n 𝑡𝑛 = 𝑆𝑛 − 𝑆𝑛−1 Arithmetic Series until the term n Arithmetic series until term before n eg. Find the value of the term at position 20 of the sequence 14, 17, 20, 23,…. 𝑡𝑛 = 𝑆𝑛 − 𝑆𝑛−1 𝑡20 = 𝑆20 − 𝑆19 = 850 − 779 = 71 20 2 𝑆20 = 28 + 20 − 1 3 = 10(28 + 19 × 3 = 10 28 + 57 = 10 × 85 = 850 19 2 𝑆19 = 28 + 19 − 1 3 = 9.5(28 + 18 × 3 = 9.5 (28 + 54) = 9.5(82) = 779 NOW DO Chapter 5 Exercise 5B Questions 1, 4, 6, 7, 8, 9a, 11, 12a