Analysis and Approaches HL Practice Paper Book | Page 21

� � � b , where the ( r �1 ) r�0 r
� Properties of the combination coefficient
�n� n!
1 . � � �r � r!( n � r)!
2 .
� n�
� n�
� �
�� �
�1
� 0�
� n�
3 .
� n�
� n
� �
��
�n
�1�
� n
�1�
4 .
n n n( n
1 )
� r �
� n
� r
( n r
1 ) r!
�n� � �: �r
� The binomial theorem : �n� � n� � n� � n � � n�
( a � b)
� � �a b � � �a b � � �a b � � � �a b � � �a b �0� �1� � 2� � n�1� � n�
n n 0 n�1 1 n�2 2 1 n�1 0 n
�n� a � � n n�r r
� � � b , where the ( r �1 ) r�0 r
� Extended binomial theorem for x � 1:
�n� -th term � �a �r
� n � n�
2 � n�
3
( 1 � x) �1� nx � � � x � � � x � � 2�
� 3�
n ( n)( n �1 ) 2 ( n)( n �1 )( n � 2 ) 3
( 1 � x) �1� nx � x � x � ( 2 )( 1 ) ( 3 )( 2 )( 1 )
n�r b r

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Mathematical Induction
� Steps of proving by mathematical induction : 1 . Prove that the statement Pn ( ) is true when n � 1
2 . Assume that Pn ( ) is true when n
� k 3 . Prove that the statement Pn ( ) is true when n�k�
1 4 . Conclude that Pn ( ) is true for all positive integer n
� Types of mathematical induction : 1 . General case 2 . Divisibility www . seprodstore . com
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