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Chapter 11: Problem 15
Decide whether each sequence is finite or infinite. The sequence of days of the week
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A briefcase has 2 locks. The combination to each lock consists of a 3-digit number, where digits may be repeated. How many combinations are possible? (Hint: The word combination is a misnomer. Lock combinations are permutations where the arrangement of the numbers is important.)
To win the jackpot in a lottery game, a person must pick 4 numbers from 0 to 9 in the correct order. If a number can be repeated, how many ways are there to play the game?
Prove statement for positive integers \(n\) and \(r,\) with \(r \leq n\). (Hint: Use the definitions of permutations and combinations.) $$\left(\begin{array}{c}n \\\n-1\end{array}\right)=n$$
It can be shown that $$(1+x)^{n}=1+n x+\frac{n(n-1)}{2 !} x^{2}+\frac{n(n-1)(n-2)}{3 !} x^{3}+\cdots$$ is true for any real number \(n\) (not just positive integer values) and any real number \(x\), where \(|x| < 1 .\) Use this series to approximate the given number to the nearest thousandth. $$(1.01)^{1.5}$$
Use the sequence feature of a graphing calculator to graph the first ten terms of each sequence as defined. Use the graph to make a conjecture as to whether the sequence converges or diverges. If you think it converges, determine the number to which it converges. $$a_{n}=\left(1+\frac{1}{n}\right)^{n}$$
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