Analysis and Approaches for IBDP Maths Ebook 2 | Page 61

18
Paper 1 Section A – Proving Divisibilities
Example
Prove by mathematical induction that 27 n �
� 1 is divisible by 13 for all n� .
[7]
Solution
When n � 1,
1
27 �1�
26
1
27 �1� 13(2)
A1
Thus, the statement is true when n � 1.
Assume that the statement is true when n� k. M1
k
27 �1� 13M
, where M � .
When n�k� 1,
k 1
k
27 1 27(27 ) 1
� � � � M1
k 1
27 1 27(13 M 1) 1
� � � � � A1
k 1
27 1 351M
26
� � � � M1
� � � � , where 27M �2� . A1
k 1
27 1 13(27M
2)
Thus, the statement is true when n�k� 1.
�
Therefore, the statement is true for all n� . R1
[7]
5
Exercise 18
1. Prove by mathematical induction that
1
9 n� � 11 is divisible by 4 for all n
�
� .
n n
�
2. Prove by mathematical induction that 9 � 4 is divisible by 5 for all n� .
[7]
[7]
3. Prove by mathematical induction that
n
3
�
� 4n
is divisible by 3 for all n� .
[7]
n
�
4. Prove by mathematical induction that 16 �12n
� 8 is divisible by 9 for all n� .
[7]
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