MATH110 December 2012
• 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 (c)
Prove the statement labelled Fact 2 which you wrote down in part a).
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!
Recall that Fact 2 is: "If for n = k,
then for ,
To prove this statement, you can assume the first part (the "if"). Starting from the "if" equation, what can you do to make it look like the "then" equation?
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As suggested in the hint, we can assume
"For n = k, ." (Equation 1)
We want to then prove that this claim also holds for , meaning that we eventually want to show that
using the previous statement (1) as our starting point. So let's start with this equation:
And in an attempt to reach equation (2), we add to both sides.
Now the left side of the equation matches (2), but what about the right hand side? Let us combine the terms to see that
This is exactly the equation what we were trying to show. Hence we have proved that if the claim holds for , it also holds for , proving Fact 2.
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