Edexcel Further Pure 2 Chapter 2 – Series: Sum simple finite series using the method of differences when the differences do not involve fractions the differences involve fractions which are given you will use partial fractions to establish the difference.

Chapter 2 – Series: FP1 Recap The following standard results from FP1 can be proved using the method of difference:

Chapter 2 – Series: Method of Differences If the general term, Un, of a series can be expressed in the form then so Then adding

Chapter 2 – Series: Method of Differences Example 1: Show that 4r3 = r2(r + 1)2 – (r - 1)2r2 Hence prove, by the method of differences that

Chapter 2 – Series: Method of Differences

Exercise 2A, Page 16 Use the method of differences to answer questions 1 and 2.

Chapter 2 – Series: Partial Fractions ∑ r = 1 (r + 1)(r + 2) 1 By using partial fraction find (r + 1)(r + 2) 1 (r + 1) A (r + 2) B = +  A = 1 and B = -1 n ∑ r = 1 (r + 1)(r + 2) 1 n ∑ r = 1 (r + 1) 1 - (r + 2) =

… ∑ n (r + 1) 1 - (r + 2) 2 1 - 3 = 3 1 - 4 + 4 1 - 5 + n 1 - n+1 +

∑ ∑ ∑ n (r + 1)(r + 2) 1 2 1 - n + 2 = 2(n + 2) (n + 2) - 2 = n Try this:

Exercise 2B, Page 16 Answer the following questions: Questions 3 and 4. Extension Task: Question 5.

Exam Questions 1. (a) Express in partial fractions. (2) (b) Hence prove, by the method of differences, that = (5) (Total 7 marks)

Exam Answers 1. (a) and attempt to find A and B M1 A1 (2)

Exam Answers (b) M1A1 [If A and B incorrect, allow A1 ft here only, providing still differences] = A1 Forming single fraction: M1 Deriving given answer : A1 (Total 7 marks)