Exam 11: Further Topics in Algebra

solve each problem involving counting theory. -A child's card game consists of a deck of 42 cards, with numbers 5 through 14, in each of four colors (green, red, black, and yellow), a red 1, and a special card called the Rook card. One card is drawn. (a) Find the probability of drawing a 5. (b) Find the probability of drawing a green card lower than 8 . (c) Find the probability of drawing the Rook card or a 10 (d) What are the odds in favor of drawing either a green or red 10 ?

Find the sum of the first nine terms of the sequence described. (a) Arithmetic with $a_{1}=18$ and $d=-3$ . (b) Geometric with $a_{1}=2$ and $r=3$ .

solve each problem involving counting theory. -Your compact disc collection consists of 9 rock, 6 jazz, and 3 classical discs. How many different ways can you play one rock, one jazz, and one classical recording?

In each sequence defined, find $a_{5}$ . (a) An arithmetic sequence with $a_{1}=9$ and $a_{3}=-11$ . (b) A geometric sequence with $a_{1}=-4$ and $r=\frac{1}{3}$ .

solve each problem involving counting theory. -An experiment consists of rolling a die four times. Find the probability of each event. (a) Exactly 2 rolls result in a 4 . (b) All four rolls result in a 5 .

solve each problem involving counting theory. -Two decks of standard playing cards, including 4 jokers, have a total of 108 cards. One card is drawn. (a) Find the probability of drawing a queen. (b) Find the probability of drawing a joker or a two. (c) Find the probability of drawing a face card (Jack, Queen, King) or a club. (d) What are the odds in favor of drawing the ace of spades?

Use mathematical induction to prove that for all positive integers $n, 7+13+19+25+\cdots+(6 n+1)=3 n^{2}+4 n$ .

Evaluate the following. (a) (b) $\left(\begin{array}{c}10 \\ 7\end{array}\right)$ (c) 10 ! (d) $P(6,1)$

Evaluate the following. (a) (b) $\left(\begin{array}{l}7 \\ 3\end{array}\right)$ (c) 9 ! (d) $P(6,4)$

Write the first four terms for each sequence. State whether the sequence is arithmetic, geometric, or neither. (a) $a_{n}=3 n-7$ (b) $a_{n}=\left(-\frac{3}{5}\right)^{n-1}$ (c) $a_{1}=4, a_{2}=6, a_{n}=2 a_{n-1}+a_{n-2}$ , for $n \geq 3$

Use mathematical induction to prove that for all positive integers $n, 5+7+9+11+\cdots+(2 n+3)=n^{2}+4 n$ .

Write the first four terms for each sequence. State whether the sequence is arithmetic, geometric, or neither. (a) $a_{n}=2 n^{2}+1$ (b) $a_{n}=\left(-\frac{5}{3}\right)^{n-1}$ (c) $a_{1}=4, a_{2}=1, a_{n}=2 a_{n-1}-a_{n-2}$ , for $n \geq 3$

solve each problem involving counting theory. -A rental car company offers 3 sizes of cars in 5 different colors. How many different cars are there if each car comes with either manual or automatic transmission and either a CD player or satellite radio?

Use mathematical induction to prove that for all positive integers $n, 1+3+5+7+\cdots+(2 n-1)=n^{2}$ .

Evaluate the following. (a) (b) $\left(\begin{array}{l}8 \\ 4\end{array}\right)$ (c) 11 ! (d) $P(8,7)$

In each sequence defined, find $a_{6}$ . (a) An arithmetic sequence with $a_{1}=-1$ and $a_{3}=5$ . (b) A geometric sequence with $a_{1}=6$ and $r=-\frac{1}{3}$ .

solve each problem involving counting theory. -A child's card game consists of a deck of 57 cards, with numbers 1 through 14, in each of four colors (green, red, black, and yellow), and a special card called the Rook card. One card is drawn. (a) Find the probability of drawing a 2. (b) Find the probability of drawing a red card lower than 10. (c) Find the probability of drawing the Rook card or a 14. (d) What are the odds in favor of drawing a 10 ?

Evaluate each sum that exists. (a) $\sum_{i=1}^{25}(6-5 i)$ (b) $\sum_{i=1}^{6} \frac{7}{5}(5)^{i}$ (c) $\sum_{i=1}^{\infty} 5\left(\frac{7}{5}\right)^{i}$ (d) $\sum_{i=1}^{\infty} 7\left(\frac{5}{7}\right)^{i}$

Evaluate each sum that exists. (a) $\sum_{i=1}^{25}(30-4 i)$ (b) $\sum_{i=1}^{6} \frac{2}{3}(3)^{i}$ (c) $\sum_{i=1}^{\infty} 2\left(\frac{2}{3}\right)^{i}$ (d) $\sum_{i=1}^{\infty}\left(\frac{3}{2}\right)^{i}$

solve each problem involving counting theory. -A child has a box containing 24 different colored markers. The child wants to write his first name in one color, his middle name in a second color, and his last name in a third color. In how many ways can this be done?