Deck 11: Further Topics in Algebra

Question 1

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

Answer 1

(a) $-2,0,4,10$ ; neither
(b) $1,-\frac{1}{4}, \frac{1}{16},-\frac{1}{64}$ ; geometric
(c) $2,6,10,14$ ; arithmetic

Question 2

In each sequence defined, find $a_{7}$ .
(a) An arithmetic sequence with $a_{1}=5$ and $a_{3}=-3$ .
(b) A geometric sequence with $a_{1}=-2$ and $r=\frac{1}{2}$ .

Answer 2

(a) -19
(b) $-\frac{1}{32}$

Question 3

Find the sum of the first eight terms of the sequence described.
(a) Arithmetic with $a_{1}=20$ and $d=-4$ .
(b) Geometric with $a_{1}=3$ and $r=2$ .

Answer 3

(a) 48
(b) 765

Question 4

Evaluate each sum that exists.
(a) $\sum_{i=1}^{25}(20-3 i)$
(b) $\sum_{i=1}^{6} \frac{4}{3}(3)^{i}$
(c) $\sum_{i=1}^{\infty} 3\left(\frac{5}{4}\right)^{i}$
(d) $\sum_{i=1}^{\infty} 5\left(\frac{4}{5}\right)^{i}$

Question 5

(a) Use the binomial theorem to expand $(4 x+y)^{4}$ .
(b) Find the fifth term in the expansion of $(w-3 y)^{7}$ .

Question 6

Evaluate the following.
(a)

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

Question 7

Use mathematical induction to prove that for all positive integers $n, 6+12+18+24+\cdots+6 n=3 n(n+1)$ .

Question 8

solve each problem involving counting theory.
-A girl opens her tackle box while fishing to find that she has 6 different sizes of hooks, 5 sizes of lead sinkers, and 3 sizes of bobbers. How many different fishing set-ups can she make if she uses one of each?

Question 9

solve each problem involving counting theory.
-Suppose that a jeweler wishes to make rings which contain exactly 3 different birthstones in a line. Since there are 12 possible birthstones available, how many such rings are possible?

Question 10

solve each problem involving counting theory.
-A group of 18 third-graders needs to form a 5 -student basketball team. If there are 7 girls and 11 boys in the class and the team is to consist of 2 girls and 3 boys, how many such teams are possible?

Question 11

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 ?

Question 12

solve each problem involving counting theory.
-An experiment consists of rolling a die six times. Find the probability of each event.
(a) Exactly 4 rolls result in a 2.
(b) None of the rolls results in a 3.

Question 13

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

Question 14

In each sequence defined, find $a_{6}$ .
(a) An arithmetic sequence with $a_{1}=7$ and $a_{3}=-1$ .
(b) A geometric sequence with $a_{1}=5$ and $r=-\frac{1}{2}$ .

Question 15

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 $r=3$ .

Question 16

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) $\sum_{i=1}^{\infty}\left(\frac{3}{2}\right)^{i}$

Question 17

(a) Use the binomial theorem to expand $(x-5 y)^{4}$ .
(b) Find the third term in the expansion of $(w-3 y)^{7}$ .

Question 18

Evaluate the following.
(a)

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

Question 19

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

Question 20

solve each problem involving counting theory.
-A rental car company offers 3 sizes of cars in 6 different models. How many different cars are there if each car comes with either manual or automatic transmission?

Question 21

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?

Question 22

solve each problem involving counting theory.
-A cheerleading squad consists of 6 boys and 5 girls. Six cheerleaders are to be selected to form a human pyramid. If the pyramid is made with 3 boys and 3 girls, how many ways can the six cheerleaders be selected?

Question 23

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?

Question 24

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 .

Question 25

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 $n \geq 3$

Question 26

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 $r=\frac{1}{3}$ .

Question 27

Find the sum of the first seven terms of the sequence described.
(a) Arithmetic with $a_{1}=11$ and $d=2$ .
(b) Geometric with $a_{1}=7$ and $r=-3$ .

Question 28

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) $\sum_{i=1}^{\infty} 7\left(\frac{5}{7}\right)^{i}$

Question 29

(a) Use the binomial theorem to expand $(3 x-y)^{4}$ .
(b) Find the sixth term in the expansion of $(w-3 y)^{7}$ .

Question 30

Evaluate the following.
(a)

(b) $\left(\begin{array}{l}7 \\ 4\end{array}\right)$
(c) 8 !
(d) $P(5,4)$

Question 31

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

Question 32

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?

Question 33

solve each problem involving counting theory.
-Suppose that a jeweler wishes to make rings which contain exactly 4 different birthstones in a line. Since there are 12 possible birthstones available, how many such rings are possible?

Question 34

solve each problem involving counting theory.
-Eighteen college students, 6 men and 12 women, must choose a group of 7 students to ride in a van to a school event. They intend to choose 4 women and 3 men to ride in the van. How many such groups are possible?

Question 35

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 black face card (Jack, Queen, King).
(b) Find the probability of drawing a joker or a red 7.
(c) Find the probability of drawing a face card (Jack, Queen, King) or a 3.
(d) What are the odds in favor of drawing the king of hearts?

Question 36

solve each problem involving counting theory.
-An experiment consists of rolling a die seven times. Find the probability of each event.
(a) Exactly 2 rolls result in a 6.
(b) None of the rolls results in a 6 .

Question 37

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 $n \geq 3$

Question 38

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 $r=-\frac{1}{3}$ .

Question 39

Find the sum of the first eight terms of the sequence described.
(a) Arithmetic with $a_{1}=23$ and $d=-4$ .
(b) Geometric with $a_{1}=4$ and $r=3$ .

Question 40

Evaluate each sum that exists.
(a) $\sum_{i=1}^{25}(5-4 i)$
(b) $\sum_{i=1}^{6} \frac{3}{4}(2)^{i}$
(c) $\sum_{i=1}^{\infty} 2\left(\frac{3}{4}\right)^{i}$
(d) $\sum_{i=1}^{\infty}\left(\frac{4}{3}\right)^{i}$

Question 41

(a) Use the binomial theorem to expand $(x-7 y)^{4}$ .
(b) Find the fourth term in the expansion of $(w-3 y)^{9}$ .

Question 42

Evaluate the following.
(a)

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

Question 43

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

Question 44

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?

Question 45

solve each problem involving counting theory.
-A child has a box containing 32 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?

Question 46

solve each problem involving counting theory.
-A high school soccer team must be made up of 7 seniors and 4 juniors. If 11 seniors and 9 juniors are eligible for the team, how many teams can be formed?

Question 47

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 ?

Question 48

solve each problem involving counting theory.
-An experiment consists of rolling a die four times. Find the probability of each event.
(a) Exactly 3 rolls result in a 6 .
(b) All four rolls result in a 5.