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 ${\displaystyle n}$.

Claim: ${\displaystyle 1+2+2^{2}+2^{3}+\cdots +2^{n}=2^{n+1}-1}$.

According to mathematical induction, two facts must be proven for the claim to be true for all positive integers ${\displaystyle n}$. The first is the following:

Fact 1: The claim holds for ${\displaystyle n=1}$; in other words, ${\displaystyle 1=2^{1}-1}$.

Carefully write down the second fact that must be proven for the claim to be demonstrated for all positive integers ${\displaystyle n}$.

Fact 2:

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