Analysis and Approaches for IBDP Maths Ebook 2 | Page 137

42
Paper 1 Section A – Mathematical Induction
Example
Prove by mathematical induction that f
( n
f
) ( x ) represents the
( n)
n
( �1) ( n �1)!
( x)
�
n 2
(1 � x)
�
1
f( x)
� (1 � x)
.
th
n derivative of
2
�
where n� and
[7]
Solution
When n � 1,
� 1 �
L.H.S. ��2 � (1)
3 �
�(1 � x)
�
�2
L.H.S. �
3
(1 � x)
R.H.S. �
1
( �1) (1 �1)!
1 2
� x �
(1 )
�2
R.H.S. �
(1 � x)
3
Thus, the statement is true when n � 1.
R1
Assume that the statement is true when n� k. M1
f
( k )
k
( �1) ( k �1)!
( x)
�
k 2
(1 � x)
�
When n�k� 1,
( k�1) d ( k)
f ( x) � ( f ( x))
M1
dx
� 1 �
( ) ( 1) ( 1)!( ( 2)) (1)
�(1 � x)
�
( k�1)
k
f x � � k � � k � � k�3
�
f
f
( k�1)
( k�1)
( x)
�
( x)
�
k�1
( �1) ( k� 2)( k�1)!
(1 � x)
k�3
k�1 k�1
( �1) ( k� 2)! ( �1) ( k�1�1)!
�
(1 �x) (1 �x)
k�3 k�1�
2
Thus, the statement is true when n�k� 1.
�
Therefore, the statement is true for all n� . R1
A1
A1
A1
[7]
10
www.seprodstore.com
127