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Mathematical Studies for the IB Diploma © Hodder Education 2010 2.5 Arithmetic sequences and series

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Mathematical Studies for the IB Diploma © Hodder Education 2010 Arithmetic sequences and series An arithmetic sequence is one in which there is a common difference (d) between successive terms. The sequences below are therefore arithmetic. +3 –4–4–4 d = 3 d = –4 5811141720 13951 –3 –7

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Mathematical Studies for the IB Diploma © Hodder Education 2010 Arithmetic sequences and series In general u 1 = the first term of an arithmetic sequence d = the common difference l = the last term An arithmetic sequence can therefore be written in its general form as: u 1 (u 1 + d) (u 1 + 2d) … (l – 2d) (l – d) l

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Mathematical Studies for the IB Diploma © Hodder Education 2010 Arithmetic sequences and series An arithmetic series is one in which the sum of the terms of an arithmetic sequence is found. e.g.The sequence 3 5 7 9 11 13 can be written as a series as 3 + 5 + 7 + 9 + 11 + 13 The sum of this series is therefore 48. The general form of an arithmetic series can therefore be written as: u 1 + (u 1 + d) + (u 1 + 2d) + … + (l – 2d) + (l – d) + l

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Mathematical Studies for the IB Diploma © Hodder Education 2010 Arithmetic sequences and series The formula for the sum (S n ) of an arithmetic series can be deduced as follows. S n = u 1 + (u 1 + d) + (u 1 + 2d) + … + (l – 2d) + (l – d) + l The formula can be written in reverse as: S n = l + (l – d) + (l – 2d) + … + (u 1 + 2d) + (u 1 + d) + u 1 If both formulae are added together we get: S n = u 1 + (u 1 + d) + (u 1 + 2d) + … + (l – 2d) + (l – d) + l S n = l + (l – d) + (l – 2d) + … + (u 1 + 2d) + (u 1 + d) + u 1 2S n = (u 1 + l) + (u 1 + l) + (u 1 + l) + … + (u 1 + l) + (u 1 + l) + (u 1 + l) +

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Mathematical Studies for the IB Diploma © Hodder Education 2010 Arithmetic sequences and series The sum of both formulae was seen to give 2S n = (u 1 + l) + (u 1 + l) + (u 1 + l) + … + (u 1 + l) + (u 1 + l) + (u 1 + l) Which in turn can be simplified to 2S n = n(u 1 + l) Therefore A formula for the sum of n terms of an arithmetic series is

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Mathematical Studies for the IB Diploma © Hodder Education 2010 Arithmetic sequences and series is not the only formula for the sum of an arithmetic series. Look again at the sequence given at the start of this presentation. 5 8 11 14 17 20 There are six terms. To get from the first term ‘5’, to the last term ‘20’, the common difference ‘3’ has been added five times, i.e. 5 + 5 × 3 = 20. The common difference is therefore added one less time than the number of terms. +3

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Mathematical Studies for the IB Diploma © Hodder Education 2010 Arithmetic sequences and series With the general form of an arithmetic series we get: u 1 + (u 1 + d) + (u 1 + 2d) + … + (l – 2d) + (l – d) + l As the common difference (d) is added one less time than the number of terms (n) in order to reach the last term (l), we can state the following: l = u 1 + (n – 1)d This can be used to generate an alternative formula to +d+d+d+d+d+d+d+d …

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Mathematical Studies for the IB Diploma © Hodder Education 2010 Arithmetic sequences and series By substituting l = u 1 + (n – 1)d into we get the following: Simplifying the formula gives Therefore the two formulae used for finding the sum of n terms of an arithmetic series are and

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