Science:Math Exam Resources/Courses/MATH110/December 2012/Question 09 (a)
• Q1 (a) • Q1 (b) • Q2 (a) • Q2 (b) • Q2 (c) • Q2 (d) • Q2 (e) • Q2 (f) • Q3 (a) • Q3 (b) • Q3 (c) • Q3 (d) • Q4 • Q5 • Q6 • Q7 (a) • Q7 (b) • Q8 • Q9 (a) • Q9 (b) • Q9 (c) • Q10 •
Question 09 (a)
In this question, you will prove by mathematical induction that the following claim holds for all positive integers .
According to mathematical induction, two facts must be proven for the claim to be true for all positive integers . The first is the following:
Fact 1: The claim holds for ; in other words, .
Carefully write down the second fact that must be proven for the claim to be demonstrated for all positive 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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
What does fact 2 say in the generic case? How can that be converted into something specific for this particular claim? Consider introducing a new notation,, with .
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.
Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies.
The second fact that must be proven is:
Let k be an integer. If the claim is true for (that is, ), then the statement must be true for (meaning ).
This fact basically states that if the claim holds for some integer, it is also true for the consecutive integer.