Question 1

Find the number of terms in the arithmetic sequence with the given conditions. $a _ { 1 } = - 11$ , $d = \frac { 1 } { 4 } , S = - 195$

Accepted Answer

A)  $n = 24$ 
B)  $n = 30$ 
C)  $n = 22$ 
D)  $n = 48$ 
E)  $n = 28$ 
A)  $n = 24$ 
B)  $n = 30$ 
C)  $n = 22$ 
D)  $n = 48$ 
E)  $n = 28$ A

Question 2

A man is 52 years old and a woman is 34 years old. The probability that the man will be alive in 10 years is 0.78, whereas the probability that the woman will be alive 10 years from now is 0.88. Assume that their life expectancies are unrelated. 
a) Find the probability that they will both be alive 10 years from now.
b) Find the probability that neither one will be alive 10 years from now.
c) Find the probability that at least one of the two will be alive 10 years from now.

Accepted Answer

A)  $P ( a ) = 0.1936 , P ( b ) = 0.0264 , P ( c ) = 0.7128$ 
B)  $P ( a ) = 0.6864 , P ( b ) = 0.0264 , P ( c ) = 0.6864$ 
C)  $P ( a ) = 0.3136 , P ( b ) = 0.9736 , P ( c ) = 0.2872$ 
D)  $P ( a ) = 0.0936 , P ( b ) = 0.9736 , P ( c ) = 0.2872$ 
E)  $P ( a ) = 0.6864 , P ( b ) = 0.0264 , P ( c ) = 0.2872$ 
A)  $P ( a ) = 0.1936 , P ( b ) = 0.0264 , P ( c ) = 0.7128$ 
B)  $P ( a ) = 0.6864 , P ( b ) = 0.0264 , P ( c ) = 0.6864$ 
C)  $P ( a ) = 0.3136 , P ( b ) = 0.9736 , P ( c ) = 0.2872$ 
D)  $P ( a ) = 0.0936 , P ( b ) = 0.9736 , P ( c ) = 0.2872$ 
E)  $P ( a ) = 0.6864 , P ( b ) = 0.0264 , P ( c ) = 0.2872$ E

Question 3

Find the sum. $$\sum _ { k = 1 } ^ { 4 } \left( k ^ { 2 } - 4 \right)$$

Accepted Answer

A)  $S = 20$ 
B)  $S = 14$ 
C)  $S = 13$ 
D)  $S = 18$ 
E)  $S = 26$ 
A)  $S = 20$ 
B)  $S = 14$ 
C)  $S = 13$ 
D)  $S = 18$ 
E)  $S = 26$ B

Question 4

A committee of 3 men and 2 women is to be chosen from a group of 11 men and 9 women. Determine the number of different ways of selecting the committee.

Accepted Answer

A)  5,940 
B)  3 
C)  35,640 
D)  165 
A)  5,940 
B)  3 
C)  35,640 
D)  165

Question 5

In how many different ways can a test consisting of five true-or-false questions be completed?

Accepted Answer

A)  10 
B)  25 
C)  7 
D)  32 
E)  64 
A)  10 
B)  25 
C)  7 
D)  32 
E)  64

Question 6

If ten basketball teams are in a tournament, find the number of different ways that first, second, and third place can be decided, assuming ties are not allowed.

Accepted Answer

A)  720 
B)  90 
C)  6 
D)  7,920 
E)  5,040 
A)  720 
B)  90 
C)  6 
D)  7,920 
E)  5,040

Question 7

Find the nth term and the fifth term of the geometric sequence. $216,36,6,1 , \ldots$

Accepted Answer

A)  $$6 ^ { - n } , \frac { 1 } { 36 }$$ 
B)  $6 ^ { 4 - n } , \frac { 1 } { 6 }$ 
C)  $$6 ^ { n - 4 } , \frac { 1 } { 36 }$$ 
D)  $2 ^ { 4 - n } , \frac { 1 } { 6 }$ 
E)  $6 ^ { n - 1 } , \frac { 1 } { 6 }$ 
A)  $$6 ^ { - n } , \frac { 1 } { 36 }$$ 
B)  $6 ^ { 4 - n } , \frac { 1 } { 6 }$ 
C)  $$6 ^ { n - 4 } , \frac { 1 } { 36 }$$ 
D)  $2 ^ { 4 - n } , \frac { 1 } { 6 }$ 
E)  $6 ^ { n - 1 } , \frac { 1 } { 6 }$

Question 8

Use the principle of mathematical induction to find the equivalent expression for every positive integer n. $1 + 2 \cdot 2 + 3 \cdot 2 ^ { 2 } + \ldots + n \cdot 2 ^ { n - 1 }$

Accepted Answer

A)  $1 + ( n - 1 ) 2 ^ { n }$ 
B)  $( n - 1 ) 2 ^ { n }$ 
C)  $9 + ( 5 n - 9 ) 2 ^ { n }$ 
D)  $11 ^ { n } - 1$ 
E)  $( 5 n - 9 ) 2 ^ { n }$ 
A)  $1 + ( n - 1 ) 2 ^ { n }$ 
B)  $( n - 1 ) 2 ^ { n }$ 
C)  $9 + ( 5 n - 9 ) 2 ^ { n }$ 
D)  $11 ^ { n } - 1$ 
E)  $( 5 n - 9 ) 2 ^ { n }$

Question 9

During 1990, smoking caused 418,890 deaths in the United States. Of these deaths, cardiovascular disease accounted for 179,820, cancer for 151,322, and respiratory diseases such as emphysema for 84,475. Find the probability that a smoking-related death was the result of either cancer or cardiovascular disease.

Accepted Answer

A)  0.813 
B)  0.209 
C)  0.791 
D)  0.803 
E)  0.786 
A)  0.813 
B)  0.209 
C)  0.791 
D)  0.803 
E)  0.786

Question 10

Without expanding completely, find the middle term in the expansion of the expression. $\left( x ^ { 1 / 3 } + y ^ { 1 / 3 } \right) ^ { 12 }$

Accepted Answer

A)  $1,320 x ^ { 3 } y ^ { 2 }$ 
B)  $924 x ^ { 3 } y ^ { 3 }$ 
C)  $1,320 x ^ { 3 } y ^ { 3 }$ 
D)  $924 x ^ { 2 } y ^ { 2 }$ 
E)  $1,320 x ^ { 2 } y ^ { 2 }$ 
A)  $1,320 x ^ { 3 } y ^ { 2 }$ 
B)  $924 x ^ { 3 } y ^ { 3 }$ 
C)  $1,320 x ^ { 3 } y ^ { 3 }$ 
D)  $924 x ^ { 2 } y ^ { 2 }$ 
E)  $1,320 x ^ { 2 } y ^ { 2 }$

Question 11

Express the sum in terms of n. $\sum _ { k = 1 } ^ { n } \left( k ^ { 2 } + 5 k + 5 \right)$ (Hint: Use the theorem on sums to write the sum as $\sum _ { k = 1 } ^ { n } k ^ { 2 } + 5 \sum _ { k = 1 } ^ { n } k + \sum _ { k = 1 } ^ { n } 5$ . Next, use the following formulas $$\begin{array} { l } 
1 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } + \ldots + n ^ { 2 } = \frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 } \
1 + 2 + 3 + \ldots + n = \frac { n ( n + 1 ) } { 2 }
\end{array}$$ and the theorem on the sum of a constant.)

Accepted Answer

A)  $\frac { n ^ { 3 } - 9 n ^ { 2 } + 23 n } { 3 }$ 
B)  $\frac { n ^ { 3 } + 9 n ^ { 2 } - 23 n } { 3 }$ 
C)  $$\frac { n ^ { 3 } + 9 n ^ { 2 } + 23 n } { 3 }$$ 
D)  $$\frac { n ^ { 3 } - 9 n ^ { 2 } - 23 n } { 3 }$$ 
E)  $\frac { n ^ { 3 } + 3 n ^ { 2 } + 46 n } { 3 }$ 
A)  $\frac { n ^ { 3 } - 9 n ^ { 2 } + 23 n } { 3 }$ 
B)  $\frac { n ^ { 3 } + 9 n ^ { 2 } - 23 n } { 3 }$ 
C)  $$\frac { n ^ { 3 } + 9 n ^ { 2 } + 23 n } { 3 }$$ 
D)  $$\frac { n ^ { 3 } - 9 n ^ { 2 } - 23 n } { 3 }$$ 
E)  $\frac { n ^ { 3 } + 3 n ^ { 2 } + 46 n } { 3 }$

Question 12

Use the principle of mathematical induction to find the equivalent expression for every positive integer n. $3 + 10 + 17 + \ldots + ( 7 n - 4 )$

Accepted Answer

A)  $n ( 7 n - 1 )$ 
B)  $\frac { n ( 7 n - 1 ) ^ { 2 } } { 2 }$ 
C)  $\frac { n ( 7 n - 1 ) } { 2 }$ 
D)  $\frac { n ^ { 2 } ( 9 n - 1 ) } { 6 }$ 
E)  $\frac { n ( 9 n - 1 ) } { 2 }$ 
A)  $n ( 7 n - 1 )$ 
B)  $\frac { n ( 7 n - 1 ) ^ { 2 } } { 2 }$ 
C)  $\frac { n ( 7 n - 1 ) } { 2 }$ 
D)  $\frac { n ^ { 2 } ( 9 n - 1 ) } { 6 }$ 
E)  $\frac { n ( 9 n - 1 ) } { 2 }$

Question 13

Bode's sequence, defined by $a _ { 1 } = 0.4 , a _ { k } = 0.1 \left( 3 \cdot 2 ^ { k - 2 } + 4 \right) \text { for } k \geq 2$ can be used to approximate distances of planets from the sun. These distances are measured in astronomical units, with 1 AU = 93,000,000 mi. For example, the third term corresponds to earth and the fifth term to the minor planet Ceres. Find the 5th term of the sequence.

Accepted Answer

A)  $a _ { 5 } = 4.1$ 
B)  $a _ { 5 } = 2.3$ 
C)  $a _ { 5 } = 3.2$ 
D)  $a _ { 5 } = 2.8$ 
E)  $a _ { 5 } = 6.8$ 
A)  $a _ { 5 } = 4.1$ 
B)  $a _ { 5 } = 2.3$ 
C)  $a _ { 5 } = 3.2$ 
D)  $a _ { 5 } = 2.8$ 
E)  $a _ { 5 } = 6.8$

Question 14

A pile of logs has 38 logs in the bottom layer, 37 in the second layer, 36 in the third, and so on. The top layer contains 28 logs. Find the total number of logs in the pile.

Accepted Answer

A)  363 logs 
B)  361 logs 
C)  726 logs 
D)  359 logs 
E)  365 logs 
A)  363 logs 
B)  361 logs 
C)  726 logs 
D)  359 logs 
E)  365 logs

Question 15

Find the number. $C ( 4,3 )$

Accepted Answer

A)  $C ( 4,3 ) = 18$ 
B)  $C ( 4,3 ) = 6$ 
C)  $C ( 4,3 ) = 24$ 
D)  $C ( 4,3 ) = 3$ 
E)  $C ( 4,3 ) = 4$ 
A)  $C ( 4,3 ) = 18$ 
B)  $C ( 4,3 ) = 6$ 
C)  $C ( 4,3 ) = 24$ 
D)  $C ( 4,3 ) = 3$ 
E)  $C ( 4,3 ) = 4$

Question 16

To win a state lottery game, a player must correctly select six numbers from the numbers 1 through 49. Find the total number of selections possible if a player selects only even numbers.

Accepted Answer

A)  96,909,120 
B)  60,410,085,120 
C)  134,596 
D)  6,991,908 
E)  144 
A)  96,909,120 
B)  60,410,085,120 
C)  134,596 
D)  6,991,908 
E)  144

Question 17

A committee is going to select 26 students from a pool of 800 to receive scholarships. How may ways could the students be selected if each scholarship is worth the same amount?

Accepted Answer

A)  $C ( 799,25 )$ 
B)  $800$ 
C)  $$26$$ 
D)  $C ( 800,26 )$ 
E)  $P ( 800,26 )$ 
A)  $C ( 799,25 )$ 
B)  $800$ 
C)  $$26$$ 
D)  $C ( 800,26 )$ 
E)  $P ( 800,26 )$

Question 18

Find the specified term of the arithmetic sequence that has the two given terms. $a _ { 12 } : a _ { 1 } = 7.8 , a _ { 2 } = 5.6$

Accepted Answer

A)  $a _ { 12 } = - 14.2$ 
B)  $a _ { 12 } = - 16.4$ 
C)  $a _ { 12 } = - 18.6$ 
D)  $a _ { 12 } = - 9.8$ 
E)  $a _ { 12 } = - 2.2$ 
A)  $a _ { 12 } = - 14.2$ 
B)  $a _ { 12 } = - 16.4$ 
C)  $a _ { 12 } = - 18.6$ 
D)  $a _ { 12 } = - 9.8$ 
E)  $a _ { 12 } = - 2.2$

Question 19

Find the fourth term of the recursively defined infinite sequence. $$a _ { 1 } = 110 , a _ { k + 1 } = \frac { 1 } { 2 } a _ { k }$$

Accepted Answer

A)  $a _ { 4 } = 17 \frac { 3 } { 4 }$ 
B)  $a _ { 4 } = 18 \frac { 3 } { 16 }$ 
C)  $a _ { 4 } = 13 \frac { 3 } { 4 }$ 
D)  $a _ { 4 } = \frac { 3 } { 4 }$ 
E)  $a _ { 4 } = 18 \frac { 3 } { 4 }$ 
A)  $a _ { 4 } = 17 \frac { 3 } { 4 }$ 
B)  $a _ { 4 } = 18 \frac { 3 } { 16 }$ 
C)  $a _ { 4 } = 13 \frac { 3 } { 4 }$ 
D)  $a _ { 4 } = \frac { 3 } { 4 }$ 
E)  $a _ { 4 } = 18 \frac { 3 } { 4 }$

Question 20

A bicycle rider coasts downhill, traveling 7 feet the first second. In each succeeding second, the rider travels 5 feet farther than in the preceding second. If the rider reaches the bottom of the hill in 10 seconds, find the total distance traveled.

Accepted Answer

A)  275 feet 
B)  300 feet 
C)  290 feet 
D)  295 feet 
E)  310 feet 
A)  275 feet 
B)  300 feet 
C)  290 feet 
D)  295 feet 
E)  310 feet

Find the number of terms in the arithmetic sequence with the given conditions. $a _ { 1 } = - 11$ , $d = \frac { 1 } { 4 } , S = - 195$

Find the sum. $\sum _ { k = 1 } ^ { 4 } \left( k ^ { 2 } - 4 \right)$

A committee of 3 men and 2 women is to be chosen from a group of 11 men and 9 women. Determine the number of different ways of selecting the committee.

In how many different ways can a test consisting of five true-or-false questions be completed?

If ten basketball teams are in a tournament, find the number of different ways that first, second, and third place can be decided, assuming ties are not allowed.

Find the nth term and the fifth term of the geometric sequence. $216,36,6,1 , \ldots$

Use the principle of mathematical induction to find the equivalent expression for every positive integer n. $1 + 2 \cdot 2 + 3 \cdot 2 ^ { 2 } + \ldots + n \cdot 2 ^ { n - 1 }$

Without expanding completely, find the middle term in the expansion of the expression. $\left( x ^ { 1 / 3 } + y ^ { 1 / 3 } \right) ^ { 12 }$

Use the principle of mathematical induction to find the equivalent expression for every positive integer n. $3 + 10 + 17 + \ldots + ( 7 n - 4 )$

A pile of logs has 38 logs in the bottom layer, 37 in the second layer, 36 in the third, and so on. The top layer contains 28 logs. Find the total number of logs in the pile.

Find the number. $C ( 4,3 )$

To win a state lottery game, a player must correctly select six numbers from the numbers 1 through 49. Find the total number of selections possible if a player selects only even numbers.

A committee is going to select 26 students from a pool of 800 to receive scholarships. How may ways could the students be selected if each scholarship is worth the same amount?

Find the specified term of the arithmetic sequence that has the two given terms. $a _ { 12 } : a _ { 1 } = 7.8 , a _ { 2 } = 5.6$

Find the fourth term of the recursively defined infinite sequence. $a _ { 1 } = 110 , a _ { k + 1 } = \frac { 1 } { 2 } a _ { k }$

A bicycle rider coasts downhill, traveling 7 feet the first second. In each succeeding second, the rider travels 5 feet farther than in the preceding second. If the rider reaches the bottom of the hill in 10 seconds, find the total distance traveled.

Topics From Algebra

Functions and Graphs

Polynomial and Rational Functions

Inverse, Exponential, and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Applications of Trigonometry

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Topics From Analytical Geometry

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Exam 9: Sequences, Series, and Probability

Find the number of terms in the arithmetic sequence with the given conditions. a1=−11a _ { 1 } = - 11a1​=−11 , d=14,S=−195d = \frac { 1 } { 4 } , S = - 195d=41​,S=−195

Find the sum. ∑k=14(k2−4)\sum _ { k = 1 } ^ { 4 } \left( k ^ { 2 } - 4 \right)∑k=14​(k2−4)

A committee of 3 men and 2 women is to be chosen from a group of 11 men and 9 women. Determine the number of different ways of selecting the committee.

In how many different ways can a test consisting of five true-or-false questions be completed?

If ten basketball teams are in a tournament, find the number of different ways that first, second, and third place can be decided, assuming ties are not allowed.

Find the nth term and the fifth term of the geometric sequence. 216,36,6,1,…216,36,6,1 , \ldots216,36,6,1,…

Use the principle of mathematical induction to find the equivalent expression for every positive integer n. 1+2⋅2+3⋅22+…+n⋅2n−11 + 2 \cdot 2 + 3 \cdot 2 ^ { 2 } + \ldots + n \cdot 2 ^ { n - 1 }1+2⋅2+3⋅22+…+n⋅2n−1

Without expanding completely, find the middle term in the expansion of the expression. (x1/3+y1/3)12\left( x ^ { 1 / 3 } + y ^ { 1 / 3 } \right) ^ { 12 }(x1/3+y1/3)12

Use the principle of mathematical induction to find the equivalent expression for every positive integer n. 3+10+17+…+(7n−4)3 + 10 + 17 + \ldots + ( 7 n - 4 )3+10+17+…+(7n−4)

A pile of logs has 38 logs in the bottom layer, 37 in the second layer, 36 in the third, and so on. The top layer contains 28 logs. Find the total number of logs in the pile.

Find the number. C(4,3)C ( 4,3 )C(4,3)

To win a state lottery game, a player must correctly select six numbers from the numbers 1 through 49. Find the total number of selections possible if a player selects only even numbers.

A committee is going to select 26 students from a pool of 800 to receive scholarships. How may ways could the students be selected if each scholarship is worth the same amount?

Find the specified term of the arithmetic sequence that has the two given terms. a12:a1=7.8,a2=5.6a _ { 12 } : a _ { 1 } = 7.8 , a _ { 2 } = 5.6a12​:a1​=7.8,a2​=5.6

Find the fourth term of the recursively defined infinite sequence. a1=110,ak+1=12aka _ { 1 } = 110 , a _ { k + 1 } = \frac { 1 } { 2 } a _ { k }a1​=110,ak+1​=21​ak​

A bicycle rider coasts downhill, traveling 7 feet the first second. In each succeeding second, the rider travels 5 feet farther than in the preceding second. If the rider reaches the bottom of the hill in 10 seconds, find the total distance traveled.

Topics From Algebra

Functions and Graphs

Polynomial and Rational Functions

Inverse, Exponential, and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Applications of Trigonometry

Systems of Equations and Inequalities

Topics From Analytical Geometry

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Find the number of terms in the arithmetic sequence with the given conditions. $a _ { 1 } = - 11$ , $d = \frac { 1 } { 4 } , S = - 195$

Find the sum. $\sum _ { k = 1 } ^ { 4 } \left( k ^ { 2 } - 4 \right)$

Find the nth term and the fifth term of the geometric sequence. $216,36,6,1 , \ldots$

Use the principle of mathematical induction to find the equivalent expression for every positive integer n. $1 + 2 \cdot 2 + 3 \cdot 2 ^ { 2 } + \ldots + n \cdot 2 ^ { n - 1 }$

Without expanding completely, find the middle term in the expansion of the expression. $\left( x ^ { 1 / 3 } + y ^ { 1 / 3 } \right) ^ { 12 }$

Use the principle of mathematical induction to find the equivalent expression for every positive integer n. $3 + 10 + 17 + \ldots + ( 7 n - 4 )$

Find the number. $C ( 4,3 )$

Find the specified term of the arithmetic sequence that has the two given terms. $a _ { 12 } : a _ { 1 } = 7.8 , a _ { 2 } = 5.6$

Find the fourth term of the recursively defined infinite sequence. $a _ { 1 } = 110 , a _ { k + 1 } = \frac { 1 } { 2 } a _ { k }$