Question 06 (b)
Use mathematical induction to prove the following statement:
For all , is a multiple of .
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
This is a routine induction problem. You might want to use the fact that
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
First, we prove the claim true when . In this case,
which is clearly divisible by 133. We assume the claim is true for . For , we have
Now, in the last line, the first summand is divisible by 133 and the second summand is also divisible by 133 by the induction hypothesis. Hence we must have that is divisible by 133 as required.