Science:Math Exam Resources/Courses/MATH220/December 2010/Question 06 (a)
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Question 06 (a)
Use mathematical induction to prove the following.
is true for all .
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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 .