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2.5 Arithmetic sequences and series 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. 5 8 +3 13 11 +3 9 –4 14 +3 5 –4 17 +3 +3 –3 1 –4 20 –4 d=3 –7 –4 d = –4 Mathematical Studies for the IB Diploma © Hodder Education 2010 Arithmetic sequences and series In general u1 = 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: u1 (u1 + d) (u1 + 2d) … (l – 2d) (l – d) l 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: u1 + (u1 + d) + (u1 + 2d) + … + (l – 2d) + (l – d) + l Mathematical Studies for the IB Diploma © Hodder Education 2010 Arithmetic sequences and series The formula for the sum (Sn) of an arithmetic series can be deduced as follows. Sn = u1 + (u1 + d) + (u1 + 2d) + … + (l – 2d) + (l – d) + l The formula can be written in reverse as: Sn = l + (l – d) + (l – 2d) + … + (u1 + 2d) + (u1 + d) + u1 If both formulae are added together we get: Sn = u1 + (u1 + d) + (u1 + 2d) + … + (l – 2d) + (l – d) + l + Sn = l + (l – d) + (l – 2d) + … + (u1 + 2d) + (u1 + d) + u1 2Sn = (u1 + l) + (u1 + l) + (u1 + l) + … + (u1 + l) + (u1 + l) + (u1 + l) Mathematical Studies for the IB Diploma © Hodder Education 2010 Arithmetic sequences and series The sum of both formulae was seen to give 2Sn = (u1 + l) + (u1 + l) + (u1 + l) + … + (u1 + l) + (u1 + l) + (u1 + l) Which in turn can be simplified to 2Sn = n(u1 + l) Therefore Sn n u1 l 2 A formula for the sum of n terms of an arithmetic series is n S n u1 l 2 Mathematical Studies for the IB Diploma © Hodder Education 2010 Arithmetic sequences and series n S n u1 l 2 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 +3 +3 +3 +3 +3 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. Mathematical Studies for the IB Diploma © Hodder Education 2010 Arithmetic sequences and series With the general form of an arithmetic series we get: u1 + (u1 + d) + (u1 + 2d) + … + (l – 2d) + (l – d) + l … +d +d +d +d 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 = u1 + (n – 1)d This can be used to generate an alternative formula to Sn n u1 l 2 Mathematical Studies for the IB Diploma © Hodder Education 2010 Arithmetic sequences and series By substituting l = u1 + (n – 1)d into Sn n u1 l 2 we get the following: Sn n u1 u1 (n 1)d 2 Simplifying the formula gives Sn n 2u1 (n 1)d 2 Therefore the two formulae used for finding the sum of n terms of an arithmetic series are n S n u1 l 2 and n Sn 2u1 (n 1)d 2 Mathematical Studies for the IB Diploma © Hodder Education 2010