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FP1 Chapter 5 - Series Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 3 rd March 2015

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Recap A series is just a sequence, which can be either finite or infinite. Euler introduced the symbol (capital sigma) to mean the sum of a series.

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Recap Determine the following results by explicitly writing out the elements in the sum. ? ?

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Sum of ones, integers, squares, cubes These are the four essential formulae you need to learn for this chapter (and that’s about it!): The last two are in the formula booklet, but you should memorise them anyway) Sum of first n integers Sum of first n squares Sum of first n cubes Note that: i.e. The sum of the first n cubes is the same as the square of the first n integers. ? ? ? ?

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Quickfire Triangulars! In your head... 1 + 2 + 3 +... + 10 = 55 1 + 2 + 3 +... + 99 = 4950 11 + 12 + 13 +... + 20 = 210 – 55 = 155 100 + 101 + 102 +... + 200 = 20100 – 4950 = 15150 ? ? ? ?

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Practice Use the formulae to evaluate the following. ? ? ? ? ? Bro Tip: Ensure that you use one less than the lower limit.

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Test Your Understanding Show that ?

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Breaking Up Summations Examples: ? ? ? ?

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Breaking Up Summations Examples: ? We can combine this property of summations with the previous one to break summations up. ? ? ?

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Past Paper Question Edexcel June 2013 ? ?

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Exercises Exercise 5E All questions Bonus Frostie Conundrum Given that n is even, determine 1 2 – 2 2 + 3 2 – 4 2 + 5 2 +... – n 2. Alternatively, notice we have pairs of difference of two pairs. We thus get: (3 × -1) + (7 × -1) + (11 × -1) +... + ([2n-1] × – 1) = -1(3 + 7 + 11 + [2n – 1]) The contents of the brackets are the sum of an arithmetic series (with a = 3, d = 4, and n/2 terms), and we could get the same result. ?

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