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. Prove that for all nonnegative integers . 
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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

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 
Hint 2 

Taking the square root before doing your proof by induction might make it easier. 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. 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 . 