Science:Math Exam Resources/Courses/MATH220/December 2010/Question 06 (b)
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Question 06 (b)
Use mathematical induction to prove the following.
for all non-negative integers .
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Another proof by mathematical induction! Remember to prove this you need to
1. Prove a base case.
2. Assume the claim is true for some k
3. Prove the claim is true for some k+1
Taking the square root before doing your proof by induction might make it easier.
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Taking the square roots on both sides shows that if we can prove that
then squaring this relation gives us our desired result. We show the above by induction (though a combinatorial argument would probably be much faster).
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 .