MATH220 December 2010
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Question 06 (a)
Use mathematical induction to prove the following.
is true for all .
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!
Recall the ingredients for a proof by induction:
1. Prove a base case.
2. Assume the claim is true for some
3. Prove the claim is true for
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For , let be the claim that
We prove that is true for all natural numbers by using induction. First, we prove that is true which is clear since
Now, we assume that is true for some (the induction hypothesis) and prove that is true. That is, we want to show that
Using the induction hypothesis, we have
Thus . Hence, by mathematical induction, we have that is true for all .
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