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

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  •  

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Question 09 (b)
Explain why the two facts from part a) imply that $1+2+2^{2}+2^{3}+2^{4}+2^{5}+2^{6}=2^{7}-1$.

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
What is the value of n in this part of the question? How would you work your way up from Fact 1 (where n = 1) to this value of n, using Fact 2?

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. We start with Fact 1 - we know that the claim is true for $n=1$. Then, using Fact 2, since the claim is true for $n=1$, we can say that it is also true for the next integer, $n=2$. We can then use Fact 2 again; since we just said the claim is true for $n=2$, it is also true for the next integer, $n=3$. We can continue this process until we get to $n=6$, which is the value of n shown in the statement of the question. (Note that you can prove the above statement is true simply by doing the calculations - but this does not use the inductive reasoning steps described in part a).

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