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Chapter 8: Problem 41
Find each indicated sum. $$\sum_{i=1}^{5} \frac{i !}{(i-1) !}$$
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Show that $$ \left(\begin{array}{l}n \\\r\end{array}\right)+\left(\begin{array}{c}n \\\r+1\end{array}\right)=\left(\begin{array}{l}n+1 \\\r+1 \end{array}\right) $$ Hints: $$ \begin{aligned}&(n-r) !=(n-r)(n-r-1) !\\\&(r+1) !=(r+1) r !\end{aligned} $$
Write the first three terms in each binomial expansion, expressing the result in simplified form. $$ (x+3)^{8} $$
Use the Binomial Theorem to expand and then simplify the result: \(\left(x^{2}+x+1\right)^{3}\). [ Hint: Write \(x^{2}+x+1\) as \(\left.x^{2}+(x+1)\right]\)
The president of a large company with \(10,000\) employees is considering mandatory cocaine testing for every employee. The test that would be used is \(90 \%\) accurate, meaning that it will detect \(90 \%\) of the cocaine users who are tested, and that \(90 \%\) of the nonusers will test negative. This also means that the test gives \(10 \%\) false positive. Suppose that \(1 \%\) of the employees actually use cocaine. Find the probability that someone who tests positive for cocaine use is, indeed, a user. What does this probability indicate in terms of the percentage of employees who test positive who are not actually users? Discuss these numbers in terms of the issue of mandatory drug testing. Write a paper either in favor of or against mandatory drug testing, incorporating the actual percentage accuracy for such tests.
Explain how to find the sum of the first \(n\) terms of a geometric sequence without having to add up all the terms.
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