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Chapter 10: Problem 75
What is a permutation?
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Some statements are false for the first few positive integers, but true for some positive integer \(m\) on. In these instances, you can prove \(S_{n}\) for \(n \geq m\) by showing that \(S_{m}\) is true and that \(S_{k}\) implies \(S_{k+1}\) when \(k \geq m .\) Use this extended principle of mathematical induction to prove that each statement in is true. Prove that \(2^{n} > n^{2}\) for \(n \geqq 5 .\) Show that the formula is true for \(n=5\) and then use step 2 of mathematical induction.
Make Sense? In Exercises \(66-69\), determine whether each statement makes sense or does not make sense, and explain your reasoning. When I toss a coin, the probability of getting heads or tails is 1 but the probability of getting heads and tails is 0.
Determine whether each statement makes sense or does not make sense, and explain your reasoning. Show that the sum of the first \(n\) positive odd integers, $$1+3+5+\cdots+(2 n-1)$$ is \(n^{2}\).
Determine whether each statement makes sense or does not make sense, and explain your reasoning. Rather than performing the addition, I used the formula \(S_{n}=\frac{n}{2}\left(a_{1}+a_{n}\right)\) to find the sum of the first 30 terms of the sequence \(2,4,8,16,32, \ldots\)
Show that $$ 1+2+3+\cdots+n=\frac{n(n+1)}{2} $$ is true for the given value of \(n .\) $$n=5: \text { Show that } 1+2+3+4+5=\frac{5(5+1)}{2}$$
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