Edexcel Further Pure 2 Chapter 2 Series Sum

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: FP 1 Recap The following standard results from FP 1 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 4 r 3 = r 2(r + 1)2 – (r - 1)2 r 2 Hence prove, by the method of differences that

Chapter 2 – Series: Method of Differences

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

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

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

n 1 ∑ r=1 = (r + 1)(r + 2) = n 1 ∑ r=1 = (r + 1)(r + 2) Try this: n ∑ r=1 1 1 - n+2 2 (n + 2) - 2 2(n + 2) n 2(n + 2) 2 (r + 1)(r + 3)

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

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

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

Exam Answers (b) M 1 A 1 [If A and B incorrect, allow A 1 ft here only, providing still differences] = A 1 Forming single fraction: M 1 Deriving given answer : A 1 (Total 7 marks)