Science:Math Exam Resources/Courses/MATH220/December 2009/Question 05 (b)
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Question 05 (b) 

Use mathematical induction to prove the following statements: For all nonnegative integers n, is a multiple of 3. 
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! 
Hint 

The proof writes itself after you set it up correctly. Remember that to prove a statement is true for all nonnegative integers , you must do three things:

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.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Let be the statement that is a multiple of 3. We prove this statement is true for all nonnegative integers using mathematical induction. Notice that is true since Now, we assume that is true for some integer and show that is true. When , we can start with the induction hypothesis for some integer . The induction hypothesis is equivalent to . To see that is true, we then notice that which shows that is true. Hence is true for all nonnegative integers by the principle of mathematical induction. 