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Chapter 8: Problem 3
Write the first six terms of each arithmetic sequence. $$a_{1}=-7, d=4$$
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Which one of the following is true? a. \(\frac{n !}{(n-1) !}=\frac{1}{n-1}\) b. The Fibonacei sequence \(1,1,2,3,5,8,13,21,34,55,89\) \(144, \ldots\) can be defined recursively using \(a_{0}=1, a_{1}=1\) \(a_{n}=a_{n-2}+a_{n-1},\) where \(n \geq 2\). c. \(\sum_{i=1}^{2}(-1)^{i} 2^{i}=0\) d. \(\sum_{i=1}^{2} a_{i} b_{i}=\sum_{i=1}^{2} a_{i} \sum_{i=1}^{2} b_{i}\)
Explain how to find the general term of a geometric sequence.
If \(f(x)=x^{4},\) find \(\frac{f(x+h)-f(x)}{h}\) and simplify.
Write the first three terms in each binomial expansion, expressing the result in simplified form. $$ \left(x^{2}+1\right)^{17} $$
a. If two people are selected at random, the probability that they do not have the same birthday (day and month) is \(\frac{255}{365} \cdot \frac{364}{368} .\) Explain why this is so. (Ignore leap years and assume 365 days in a year.) b. If three people are selected at random, find the probability that they all have different birthdays. c. If three people are selected at random, find the probability that at least two of them have the same birthday. d. If 20 people are selected at random, find the probability that at least 2 of them have the same birthday. e. How large a group is needed to give a 0.5 chance of at least two people having the same birthday?
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