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
n
By using partial fraction find
1
∑
r = 1 (r + 1)(r + 2)
1
=
n
∑
+
(r + 1)
(r + 1)(r + 2)

A
B
(r + 2)
A = 1 and B = -1
1
r = 1 (r + 1)(r + 2)
n
=
∑
1
r = 1 (r + 1)
1
(r + 2)
n
∑
1
r = 1 (r + 1)
-
1
=
(r + 2)
+
…
+
+
+
1
-
1
2
3
1
1
-
3
4
1
1
-
4
5
1
1
-
n
n+1
1
1
n+1
-
n+2
=
1
2
-
1
n+2
n
∑
1
1
=
r = 1 (r + 1)(r + 2)
-
n+2
2
=
1
(n + 2) - 2
2(n + 2)
n
∑
1
=
2(n + 2)
r = 1 (r + 1)(r + 2)
Try this:
n
n
∑
2
r = 1 (r + 1)(r + 3)
Exercise 2B, Page 16

Answer the following questions:
Questions 3 and 4.

Extension Task:
Question 5.
Exam Questions
1.
(a)
Express
(b)
Hence prove, by the method of differences, that
n

r 1
1
in partial fractions.
r (r  2)
4
r (r  2)
=
n(3n  5)
(n  1)( n  2)
(2)
(5)
(Total 7 marks)
Exam Answers
1.
(a)
A(r  2)  Br
1
A
B
 

r (r  2) r r  2
r (r  2)
and attempt to find A and B
M1
1
1


2r 2(r  2)
A1
(2)
Exam Answers
(b)

n

1
4
1 
1
 2 

r (r  2)
r r  2
1 
1  1 1  1 1 
1
 r  r  2   {1  3}   2  4    3  5   .......



 

1  1
1 
 1




 

 n  1 n  1  n n  2 
M1A1
[If A and B incorrect, allow A1 ft here only, providing still differences]
3
1
1


=
2 n 1 n  2
Forming single fraction:
Deriving given answer :
A1
3(n  1)(n  2)  2(n  2)  2(n  1)
2(n  1)(n  2)
n(3n  5)
(n  1)(n  2)
M1
A1
(Total 7 marks)