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

MATH110 December 2012

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Question 09 (a)

In this question, you will prove by mathematical induction that the following claim holds for all positive integers $n$. Claim: $1+2+2^{2}+2^{3}+\cdots +2^{n}=2^{n+1}-1$. According to mathematical induction, two facts must be proven for the claim to be true for all positive integers $n$. The first is the following: Fact 1: The claim holds for $n=1$; in other words, $1=2^{1}-1$. Carefully write down the second fact that must be proven for the claim to be demonstrated for all positive integers $n$. Fact 2:

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 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,$k$, with $k<n$.

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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. The second fact that must be proven is: Let k be an integer. If the claim is true for $n=k$ (that is, $2^{1}+2^{2}+\dots +2^{k}=2^{k+1}-1$), then the statement must be true for $n=k+1$ (meaning $2^{1}+2^{2}+\dots +2^{k}+2^{k+1}=2^{k+2}-1$). This fact basically states that if the claim holds for some integer, it is also true for the consecutive integer.

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