Science:Math Exam Resources/Courses/MATH110/December 2012/Question 09 (c)

MATH110 December 2012

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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!

Hint
Recall that Fact 2 is: "If for n = k, $2^{1}+2^{2}+\dots +2^{k}=2^{k+1}-1$ then for $n=k+1$, $2^{1}+2^{2}+\dots +2^{k+1}=2^{k+2}-1$. 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?

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. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. As suggested in the hint, we can assume "For n = k, $2^{1}+2^{2}+\dots +2^{k}=2^{k+1}-1$." (Equation 1) We want to then prove that this claim also holds for $n=k+1$, meaning that we eventually want to show that $2^{1}+2^{2}+\dots +2^{k+1}=2^{k+2}-1$ (Equation 2) using the previous statement (1) as our starting point. So let's start with this equation: $2^{1}+2^{2}+\dots +2^{k}=2^{k+1}-1$. And in an attempt to reach equation (2), we add $2^{k+1}$ to both sides. $2^{1}+2^{2}+\dots +2^{k}+2^{k+1}=2^{k+1}-1+2^{k+1}$. Now the left side of the equation matches (2), but what about the right hand side? Let us combine the terms to see that ${\begin{aligned}2^{1}+2^{2}+\dots +2^{k}+2^{k+1}&=2^{k+1}+2^{k+1}-1\\&=2(2^{k+1})-1\\&=2^{k+1+1}-1\\&=2^{k+2}-1\\\end{aligned}}$. This is exactly the equation what we were trying to show. Hence we have proved that if the claim holds for $n=k$, it also holds for $n=k+1$, proving Fact 2.

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