Analysis and Approaches for IBDP Maths Ebook 1 | Page 85

Front Page
29
Paper 1 Section A – Proofs of number properties
Example
(a)
(b)
2 2
(2n 1) (2 n) 4n
1
Show that � � � � , where n� .
[2]
Hence, or otherwise, prove that the difference of the squares of an odd integer and
its consecutive smaller even integer is odd.
[3]
Solution
(a)
L.H.S.
� (2n�1) � (2 n)
2 2
2 2
� � � � M1A1
4n 4n 1 4n
�4n
� 1
�R.H.S. AG N0
(b) 2n � 1 is an odd integer and 2n is an even integer,
and they are consecutive integers.
R1
2 2
(2n �1) �(2 n) � 4n
� 1
A1
Also 4n � 1 is an odd integer.
R1
Thus, the difference of the squares of an odd
integer and its consecutive smaller even integer
is odd. AG N0
[2]
[3]
8
Exercise 29
1. (a) Show that
(b)
2 2 2
(3 n) (3n 3) 18n 18n
9
� � � � � , where n� .
Hence, or otherwise, prove that the sum of the squares of any two consecutive
multiples of 3 is odd.
[2]
[3]
2. (a) Show that 2 n � 1 1
2 � �
2n�1 2n� 1
, where n� . [2]
(b)
Hence, or otherwise, prove that the ratio of any odd integer to its consecutive
smaller odd integer is not equal to 1.
[3]
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