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ARITHMETIC SEQUENCES AND SERIES Week Commencing Monday 5th October Learning Intention: • To be able to find the sum of an arithmetic series Contents: 1. Sum of an Arithmetic Series Formula 2. Sum Formula Proof 3. Using the Sum Formula 4. Real-Life Problems 5. Assignment 4 ARITHMETIC SEQUENCES AND SERIES Sum of an Arithmetic Series The sum of the first n terms of a series is generally denoted by Sn. For arithmetic series there is a formula to work out the sum of the first n terms: Sn n 2 2a (n -1)d where n is the number of terms a is the first term d is the common difference ARITHMETIC SEQUENCES AND SERIES Proof for Sum of an Arithmetic Series You need to be able to reproduce this proof for the Core 1 examination. Sn a (a d) (a 2d) ... (a (n -1)d) This can be written in reverse : Sn (a (n -1)d) (a (n - 2)d) ... (a d) a Adding the two sums together we get : 2Sn (2a (n -1)d) (2a (n -1)d) ... (2a (n -1)d) On the RHS we now have n lots of (2a (n -1)d). Therefore we have : 2Sn n[2a (n -1)d] Dividing across be 2 we get : Sn n 2 [2a (n 1)d] ARITHMETIC SEQUENCES AND SERIES Using the Sum Formula Example: Find the sum of the series 4 + 9 + 14 + 19 + … to 20 terms Solution: First term, a = 4 Common Difference, d = 5 Number of terms, n = 20 Substitute into formula: Sn S20 20 2 2(4) (20 -1)5 S20 10[8 19(5)] S20 10[8 95] S20 10[103] S20 1030 n 2 2a (n -1)d ARITHMETIC SEQUENCES AND SERIES Using the Sum Formula Example: Find the sum of the series 2 + 6 + 10 + 14 + … + 158 Solution: First term, a = 2 Common Difference, d = 4 Number of terms, n = ? We need to find n before we can find the sum. Un = 158 Un = a + (n -1)d 158 = 2 + (n – 1)(4) 158 = 2 + 4n – 4 158 = 4n – 2 160 = 4n 40 = n cont’d on next slide ARITHMETIC SEQUENCES AND SERIES Using the Sum Formula Example Continued: Find the sum of the series 2 + 6 + 10 + 14 + … + 158 Solution Continued: First term, a = 2 Common Difference, d = 4 Number of terms, n = 40 We can now substitute into the sum formula to give: S40 40 2 2(2) (40 -1)4 S40 20[4 39(4)] S40 20[4 156] S40 20[160] S40 3200 ARITHMETIC SEQUENCES AND SERIES Real-Life Problems Arithmetic Series are used in everyday life to solve problems. Example: Ahmed plans to save £250 in the year 2001, £300 in 2002, £350 in 2003, and so on until the year 2020. Is savings for and arithmetic series with common difference £50. a. Find the amount he plans to save in 2011 b. Calculate his total planned savings over the 20-year period from 2001 to 2020. Solution on next slide. ARITHMETIC SEQUENCES AND SERIES Real-Life Problems Solution: a. a = £250 d = £50 From 2001 to 2011 is 11 years which means we want the 11th term in the series. 2011: £250 + (11 – 1)(£50) = £250 + 10(£50) = £250 + £500 = £750 b. a = £250 S20 20 2 d = £50 2(£250) (20 -1)£50 S20 10[£500 19(£50)] S20 10[£500 £950] S20 10[£1450] S20 £14 500 n = 20 ARITHMETIC SEQUENCES AND SERIES Assignment 4 – Sum of Arithmetic Series Follow the link for Assignment 4 on Sum of Arithmetic Series in the Moodle Course Area. This is a Yacapaca Activity. Completed assignments must be submitted by 5:00pm on Monday 12th October.