Analysis and Approaches for IBDP Maths Ebook 2 | Page 59

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Paper 1 Section A – General Cases
Example
Prove by mathematical induction that
n
�
r�1
n( n �1)(2n
�5)
�
rr ( �2)
�
, n� .
6
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Solution
5
When n � 1,
L.H.S. � rr ( �2)
1
�
r�1
L.H.S. �� 1
1(1 �1)(2(1) �5)
R.H.S. �
6
R.H.S. �� 1
Thus, the statement is true when n � 1.
R1
Assume that the statement is true when n� k. M1
k
k( k �1)(2k
�5)
� rr ( �2)
�
6
r�1
When n�k� 1,
k�1
�r( r � 2) � � r( r � 2) � ( k �1)( k �1�
2)
M1
r�1 r�1
r�1
k
k�1
k( k �1)(2k �5) 6( k �1)( k �1)
� rr ( � 2) � �
A1
6 6
k�1
� r( r � 2) � �k(2k �5) � 6( k �1)
�
A1
r�1
k�1
�
r�1
r�1
k �1
6
k �1
r r k k
6
2
( � 2) � (2 � � 6)
k�1
( k �1)( k � 2)(2k
�3)
� rr ( �2)
�
A1
6
k�1
( k �1)(( k �1) �1)(2( k �1) �5)
� rr ( �2)
�
A1
6
r�1
Thus, the statement is true when n�k� 1.
�
Therefore, the statement is true for all n� . R1
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