Science:Math Exam Resources/Courses/MATH220/December 2009/Question 05 (c)
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Question 05 (c)
Use mathematical induction to prove the following statements:
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!
The proof writes itself after you set it up correctly. Remember to apply mathematical induction on a statement you must do three things:
1. Prove is true
2. Assume is true for some natural number k
3. Show that is true based on the assumption above.
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Let be the statement that
We prove this statement true for all natural numbers using mathematical induction. Notice that is true since
Now, we assume that is true for some and show that is true. When , we can start with the induction hypothesis:
To show is true, notice that :
which shows that is true. Hence is true for all natural numbers by the principle of mathematical induction.