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© Boardworks Ltd of 37 © Boardworks Ltd of 37 AS-Level Maths: Core 1 for Edexcel C1.6 Sequences and series This icon indicates the slide contains activities created in Flash. These activities are not editable. For more detailed instructions, see the Getting Started presentation.

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© Boardworks Ltd of 37 Contents © Boardworks Ltd of 37 Sequences The formula for the n th term Recurrence relations Arithmetic sequences Arithmetic series The sum of the first n natural numbers The sum of an arithmetic series Using Σ notation Examination-style questions Sequences

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© Boardworks Ltd of 37 Sequences In mathematics, a sequence is a succession of numbers, called terms, that follow a given rule. For example: 9, 16, 25, 36, 49, … is a sequence of square numbers starting with 9. A sequence can be infinite, as shown by the … at the end of the sequence shown above, or it can be finite. For example: A sequence can be defined by: a formula for the n th term of the sequence, or a recurrence relation together with the first term of the sequence. 3, 6, 12, 24, 48, 96 is a finite sequence containing six terms.

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© Boardworks Ltd of 37 Contents © Boardworks Ltd of 37 Sequences The formula for the n th term Recurrence relations Arithmetic sequences Arithmetic series The sum of the first n natural numbers The sum of an arithmetic series Using Σ notation Examination-style questions The formula for the n th term

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© Boardworks Ltd of 37 The formula for the n th term The n th term, or the general term, of a sequence is often given using superscript (or suffix) notation as u n. the 2 nd term is u 2, the 3 rd term is u 3, the 4 th term is u 4, The 1 st term is then called u 1, Any term in a sequence can be found by substituting its position number into a given formula for u n. the 5 th term is u 5 and so on. Letters other than u can be used. For example, the terms in a sequence could also be given by t 1, t 2, t 3, t 4, … t n.

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© Boardworks Ltd of 37 The formula for the n th term For example, the formula for the n th term of a sequence is given by u n = 4 n – 5. u 1 = 4 × 1 – 5 =–1 u 2 = 4 × 2 – 5 =3 u 3 = 4 × 3 – 5 =7 u 4 = 4 × 4 – 5 =11 u 5 = 4 × 5 – 5 =15 The first five terms in the sequence are: –1, 3, 7, 11 and 15. Find the first five terms in the sequence.

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© Boardworks Ltd of 37 The formula for the n th term

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© Boardworks Ltd of 37 Contents © Boardworks Ltd of 37 Sequences The formula for the n th term Recurrence relations Arithmetic sequences Arithmetic series The sum of the first n natural numbers The sum of an arithmetic series Using Σ notation Examination-style questions Recurrence relations

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© Boardworks Ltd of 37 Recurrence relations This sequence can also be defined by a recurrence relation. To define a sequence using a recurrence relation we need the value of the first term and an expression relating each term to a previous term. u 1 = –1 u 2 = u = 3 u 3 = u = 7 u 4 = u = 11 and so on. For the sequence –1, 3, 7, 11, 15, …, each term can be found by adding 4 to the previous term. We can write: In general: u n+1 = u n + 4

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© Boardworks Ltd of 37 Recurrence relations A recurrence relation together with the first term of a sequence is called an inductive definition. So the inductive definition for the sequence –1, 3, 7, 11, 15, … is u 1 = –1, u n +1 = u n + 4. u 1 = 3 u 2 = (2 × 3) + 1 =7 u 3 = (2 × 7) + 1 =15 u 4 = (2 × 15) + 1 =31 u 5 = (2 × 31) + 1 =63 So the first five terms in the sequence are 3, 7, 15, 31 and 63. A sequence is given by the recurrence relation u n +1 = 2 u n + 1 with u 1 = 3. Write down the first five terms of the sequence.

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© Boardworks Ltd of 37 Using an inductive definition

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© Boardworks Ltd of 37 Contents © Boardworks Ltd of 37 Sequences The formula for the n th term Recurrence relations Arithmetic sequences Arithmetic series The sum of the first n natural numbers The sum of an arithmetic series Using Σ notation Examination-style questions Arithmetic sequences

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© Boardworks Ltd of 37 Arithmetic sequences In an arithmetic sequence (or arithmetic progression) the difference between any two consecutive terms is always the same. This is called the common difference. For example, the sequence: 8, 11, 14, 17, 20, … is an arithmetic sequence with 3 as the common difference. We could write this sequence as: 8,8 + 3, , , , … or 8,8 + 3,8 + (2 × 3),8 (3 × 3), 8 + (4 × 3), …

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© Boardworks Ltd of 37 Arithmetic sequences If we call the first term of an arithmetic sequence a and the common difference d we can write a general arithmetic sequence as: a,a, a + d, a + 2 d, a + 3 d, a + 4 d, … Also: The n th term of an arithmetic sequence with first term a and common difference d is a + ( n – 1) d The inductive definition of an arithmetic sequence with first term a and common difference d is u 1 = a, u n +1 = u n + d

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© Boardworks Ltd of 37 Arithmetic sequences This is an arithmetic sequence with first term a = 10 and common difference d = –3. The n th term is given by a + ( n – 1) d so: u n = 10 – 3( n – 1) = 10 – 3 n + 3 = 13 – 3 n u 1 = 13 – 3 × 1 = 10 u 3 = 13 – 3 × 3 = 4 u 2 = 13 – 3 × 2 = 7 Let’s check this formula for the first few terms in the sequence: What is the formula for the n th term of the sequence 10, 7, 4, 1, –2 …?

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© Boardworks Ltd of 37 Arithmetic sequences This is an arithmetic sequence with first term a = –7 and common difference d = 6. The n th term is given by a + ( n – 1) d so: u n = –7 + 6( n – 1) = –7 + 6 n – 6 = 6 n – 13 We can find the value of n for the last term by solving: 6 n – 13 = 71 6 n = 84 n = 14 So, there are 14 terms in the sequence. Find the number of terms in the finite arithmetic sequence –7, –1, 5, … 71.

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© Boardworks Ltd of 37 Arithmetic sequences Using the 4 th term: a + 3 d = 12 Using the 20 th term: a + 19 d = 92 Subtracting the first equation from the second equation gives: 16 d = 80 d = 5 Substitute this into the first equation: a + 15 = 12 a = –3 The n th term of an arithmetic sequence with a = –3 and d = 5 is: u n = –3 + 5( n –1) = –3 + 5 n – 5 = 5 n – 8 The 4 th term in an arithmetic sequence is 12 and the 20 th term is 92. What is the formula for the n th term of this sequence?

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© Boardworks Ltd of 37 Contents © Boardworks Ltd of 37 Sequences The formula for the n th term Recurrence relations Arithmetic sequences Arithmetic series The sum of the first n natural numbers The sum of an arithmetic series Using Σ notation Examination-style questions Arithmetic series

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© Boardworks Ltd of 37 Series The sum of all the terms of a sequence is called a series. 1, 3, 5, 7, 9, … is a sequence while: … is a series. For example: When the difference between each term in a series is constant, as in this example, the series is called an arithmetic series or arithmetic progression (AP for short). The sum of a series containing n terms is often denoted by S n, so for the series given above we could write: S 5 = = 25 When n is large, a more systematic approach for calculating the sum of a given number of terms is required.

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© Boardworks Ltd of 37 Gauss’ method It is said that when the famous mathematician Karl Friedrich Gauss was a young boy at school, his teacher asked the class to add together every whole number from one to a hundred. The teacher expected this activity to keep the class occupied for some time and so he was amazed when Gauss put up his hand and gave the answer, 5050, almost immediately!

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© Boardworks Ltd of 37 Gauss’ method Gauss worked the answer out by noticing that you can quickly add together consecutive numbers by writing the numbers once in order and once in reverse order and adding them together. So to add the numbers from 1 to 100: … … S = … S = 2 S = So:2 S = 100 × 101 = S = 5050

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© Boardworks Ltd of 37 Contents © Boardworks Ltd of 37 Sequences The formula for the n th term Recurrence relations Arithmetic sequences Arithmetic series The sum of the first n natural numbers The sum of an arithmetic series Using Σ notation Examination-style questions The sum of the first n natural numbers

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© Boardworks Ltd of 37 The sum of the first n natural numbers To find the sum of the first n natural numbers we can generalize Gauss’ method as follows. Write the sum of the first n natural numbers as: …+ ( n – 2) + ( n –1) + n S = n + ( n –1) + ( n – 2) +… S =( n + 1) + + +… S = This gives us: 2 S = n ( n + 1) So: The sum of the first n natural numbers is given by

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© Boardworks Ltd of 37 The sum of the first n natural numbers What is the sum of the first 30 natural numbers? … + 30 = = 465 What is the sum of the natural numbers from 21 to 30? … + 30 =( … + 30) – ( … + 20) = 465 – 210 = 255

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© Boardworks Ltd of 37 Contents © Boardworks Ltd of 37 Sequences The formula for the n th term Recurrence relations Arithmetic sequences Arithmetic series The sum of the first n natural numbers The sum of an arithmetic series Using Σ notation Examination-style questions The sum of an arithmetic series

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© Boardworks Ltd of 37 The sum of an arithmetic series Gauss’ method can be applied to any arithmetic series of the general form a + ( a + d ) + ( a + 2 d ) + ( a +3 d ) + … + ( a + ( n – 1) d ) where a is the first term in the series, d is the common difference and n is the number of terms. Let’s call the last term l so that: l = ( a + ( n – 1) d ) The sum of the first n terms can now be written as: ( a + l ) + + a +++…+ ( l – 2 d ) + ( l – d ) + l S n =( a + d )( a + 2 d ) l +++…+ + ( a + d ) + a S n =( l – d )( l – 2 d )( a + l ) +… Sn=2Sn=

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© Boardworks Ltd of 37 The sum of an arithmetic series This gives us:2 S n = n ( a + l ) So: The sum of the first n terms in an arithmetic series is where a is the first term and l is the last. If the last term is not known this formula can be written in terms of a and n by substituting ( a + ( n – 1) d ) for l in the above. An alternative formula for the sum of an arithmetic series is then:

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© Boardworks Ltd of 37 Find the sum of the first 20 terms of the arithmetic series … The sum of an arithmetic series We don’t know the last term so we can use: with a = 5, d = 6 and n = 20. S 20 = 10( ) = 1240

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© Boardworks Ltd of 37 Arithmetic series

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© Boardworks Ltd of 37 Contents © Boardworks Ltd of 37 Sequences The formula for the n th term Recurrence relations Arithmetic sequences Arithmetic series The sum of the first n natural numbers The sum of an arithmetic series Using Σ notation Examination-style questions Using Σ notation

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© Boardworks Ltd of 37 Using Σ notation When working with series, the Greek symbol Σ (the capital letter sigma) is used to mean ‘the sum of’. For example: represents a finite series containing n terms: This is the first value of r … … and this is the last value of r. u 1 + u 2 + u 3 + … + u n The terms in the series are obtained by substituting 1, 2, 3, …, n in turn for r in u r.

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© Boardworks Ltd of 37 Using Σ notation For example, suppose we want to find the sum of the first 4 terms of the series whose n th term is of the form 3 n – 1. We can write: The initial value of r doesn’t have to be 1. For example: (3 × 1 – 1) + (3 × 2 – 1) + (3 × 3 – 1) + (3 × 4 – 1) = Infinite series are given by writing an ∞ symbol above the Σ. For example:

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© Boardworks Ltd of 37 Using Σ notation

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© Boardworks Ltd of 37 Using Σ notation Evaluate Substituting r = 2, 3, 4, …,15 into 25 – 2 r gives us the arithmetic series … + –5. We can evaluate this by putting a = 21, l = –5 and n = 14 into the formula So: = 112 There are 14 terms in this sequence because both r = 2 and r = 15 are included.

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© Boardworks Ltd of 37 Contents © Boardworks Ltd of 37 Sequences The formula for the n th term Recurrence relations Arithmetic sequences Arithmetic series The sum of the first n natural numbers The sum of an arithmetic series Using Σ notation Examination-style questions

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© Boardworks Ltd of 37 Examination-style question The sum of the first 3 terms of an arithmetic series is 21 and the sum of the next three terms is 66. a) The sum of the first 3 terms can be written as: a)Find the value of the first term and the common difference. b)Write an expression for the n th term of the series u n. c)Find the sum of the first 10 terms. a + ( a + d ) + ( a + 2 d ) = 3 a + 3 d a + d = a + 3 d = 21So The sum of the next 3 terms can be written as: ( a + 3 d ) + ( a + 4 d ) + ( a + 5 d ) = 3 a + 12 d a + 4 d = a + 12 d = 66So

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© Boardworks Ltd of 37 Examination-style question 2 – : 1 3 d = 15 d = 5 a = 2 b) In general, for an arithmetic series u n = a + ( n – 1) d so u n = 2 + 5( n – 1) = 5 n – 3 = 245 c) u 10 = (5 ×10) – 3 Now using the formula with a = 2 and l = 47: = 47

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