Exam 9: Discrete Mathematics

The sequence 2,6,18,54, .... is geometric. (a) What is the common ratio of the sequence? (b) What is a recursive rule for the n th term $(n \geq 2)$ of the sequence?

Suppose there is a 40% chance of rain tomorrow. If it rains, there is a 20% chance that all of the rides at an amusement park will be operating. If it doesn't rain, there is a 90% chance all of the rides will be operating. What is the probability that all of the rides will be operating tomorrow?

A fair coin is tossed five times. Find the probability of tossing exactly two tails in those five tosses.

Questions $12-15$ refer to the following data set. $\{51,57,43$ , $65,72,39,56,61,48,49,37,44,68,75,52,56,41,63\}$ $\text { Construct a stemplot for the data. }$

Assume that the probability that a newborn child is a female is 50% . In a family of four children, what is the probability that (a) all the children are girls? (b) at least two of the children are boys?

Prove the following statement by mathematical induction. $1+7+13+\ldots+(6 n-5)=n(3 n-2)$

Which one of the following sequences is divergent?

Sketch a boxplot for the data . Stem Leaf 3 556 4 23368 5 235567 6 3444556 7 23455677

Using the binomial theorem, expand the binomial $(2 x+y)^{5}$

Use the formulas $\sum_{k=1}^{n} k=\frac{n(n+1)}{2}$ and $\sum_{k=1}^{n} k^{2}=\frac{n(n+1)(2 n+1)}{6}$ to find the sum in terms of $\text { n. } \sum_{k=1}^{n}\left(3 k^{2}-5 k+7\right)$

Determine whether the infinite geometric series converges. If it does converge, what is the sum? $\frac{5}{4}+\frac{5}{16}+\frac{5}{64}+\ldots$

Questions $12-15$ refer to the following data set. $\{51,57,43$ , $65,72,39,56,61,48,49,37,44,68,75,52,56,41,63\}$ In the space below, construct a frequency table for the data. Then draw the histogram. Use intervals of 10.

Prove the following statement by mathematical induction. $3+7+11+\ldots+(4 n-1)=n(2 n+1)$

Expand $(5 x-8 y)^{5}$

Find the sum of the infinite series $\sum_{k=1}^{\infty} 4\left(\frac{1}{5}\right)^{k-1}$

Questions $12-15$ refer to the following data set. $\{27,42,33$ , $45,52,18,37,37,29,42,58,27,19,56,45,38\}$ In the space below, construct a frequency table for the data. Then draw the histogram. Use intervals of 10 .

Find the probability of obtaining exactly six heads in ten tosses of a fair coin.

Does correlation imply causation?

The sum of the squares of the first n positive odd integers is $\frac{(2 n-1)(2 n)(2 n+1)}{6}$ Use this expression to find the sum $1^{2}+3^{2}+5^{2}+\cdots+75^{2}$

Questions $12-15$ refer to the following data set. $\{27,42,33$ , $45,52,18,37,37,29,42,58,27,19,56,45,38\}$ $\text { Construct a stemplot for the data. }$