Question 1

Use Recursion Formulas
-$a _ { 1 } = - 6$ and $a _ { n } = - 4 a _ { n - 1 }$ for $n \geq 2$

Accepted Answer

A)  $- 6,24 , - 96,384$ 
B)  $6 , - 24,96 , - 384$ 
C)  $- 6 , - 24 , - 96 , - 384$ 
D)  $- 6,26 , - 98,386$

Question 2

The general term of a sequence is given. Determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio.
-$$a _ { n } = 5 ^ { n }$$

Accepted Answer

A)  geometric, $r = 5$ 
B)  arithmetic, $\mathrm { d } = 5$ 
C)  geometric, $r = 6$ 
D)  neither

Question 3

Additional Concepts
-$\frac { { } _ { 7 } C _ { 3 } } { { } _ { 5 } C _ { 1 } } - \frac { 48 ! } { 46 ! }$

Accepted Answer

A)  $- 2249$ 
B)  2249 
C)  $- 2345$ 
D)  $- 41$

Question 4

Write Terms of an Arithmetic Sequence
-$$a _ { 1 } = - \frac { 3 } { 8 } ; d = - \frac { 1 } { 8 }$$

Accepted Answer

A)  $- \frac { 3 } { 8 } , - \frac { 1 } { 2 } , - \frac { 5 } { 8 } , - \frac { 3 } { 4 } , - \frac { 7 } { 8 }$ 
B)  $- \frac { 3 } { 8 } , - \frac { 1 } { 4 } , - \frac { 1 } { 8 } , 0 , \frac { 1 } { 8 }$ 
C)  $- \frac { 3 } { 8 } , - \frac { 3 } { 4 } , - \frac { 9 } { 8 } , - \frac { 3 } { 2 } , - \frac { 15 } { 8 }$ 
D)  $- \frac { 3 } { 8 } , - \frac { 1 } { 2 } , - \frac { 1 } { 8 } , - \frac { 3 } { 4 } , \frac { 1 } { 8 }$

Question 5

Write the word or phrase that best completes each statement or answers the question. Use mathematical induction to prove that the statement is true for every positive integer n.
-$$8 + 16 + 24 + \ldots + 8 n = 4 n ( n + 1 )$$

Accepted Answer

The answer of Write the word or phrase that best...

Question 6

Find the Probability that an Event Will Not Occur
-A bag contains 3 blue marbles, 8 green marbles, and 5 red marbles. One marble is drawn from the bag. What is the probability that the marble drawn is not blue?

Accepted Answer

A)  $\frac { 13 } { 16 }$ 
B)  $\frac { 3 } { 13 }$ 
C)  $\frac { 3 } { 16 }$ 
D)  $\frac { 13 } { 3 }$

Question 7

Probability
1 Compute Empirical Probability
-During July in Jacksonville, Florida, it is not uncommon to have afternoon thunderstorms. On average, 12.3 days have afternoon thunderstorms. What is the probability that a randomly selected day in July will not have a thunderstorm?

Accepted Answer

A)  60.32% 
B)  59.00% 
C)  39.68% 
D)  87.70%

Question 8

Use the Formula for the General Term of a Geometric Sequence
-Find a 8 when $a _ { 1 } = 8,000,000 , r = 0.1$.

Accepted Answer

A)  $0.8$ 
B)  $0.08$ 
C)  8 
D)  $0.008$

Question 9

Find the Probability of One Event and a Second Event Occurring
-A 6 -sided die is rolled. What is the probability of rolling a number that is even and a 5 ?

Accepted Answer

A)  0 
B)  $\frac { 1 } { 6 }$ 
C)  $\frac { 1 } { 2 }$ 
D)  1 6 2

Question 10

Express the sum using summation notation. Use a lower limit of summation not necessarily 1 and k for the index of summation.
-$$4 + \frac { 9 } { 2 } + 5 + \frac { 11 } { 2 } + \ldots + 9$$

Accepted Answer

A)  $\sum _ { \mathrm { k } = 8 } ^ { 18 } \frac { \mathrm { k } } { 2 }$ 
B)  $\sum _ { \mathrm { k } = 1 } ^ { 18 } \frac { \mathrm { k } } { 2 }$ 
C)  $\sum _ { \mathrm { k } = 8 } ^ { 12 } \frac { \mathrm { k } } { 2 }$ 
D)  $\sum _ { \mathrm { k } = 2 } ^ { 18 } \frac { \mathrm { k } } { 2 }$

Question 11

Use the Permutations Formula
-The matching section of an exam has 5 questions and 7 possible answers. In how many different ways can a student answer the 5 questions, if none of the answer choices can be repeated?

Accepted Answer

A)  2520 
B)  42 
C)  21 
D)  84

Question 12

Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence.
-$$\sum _ { i = 1 } ^ { 3 } \left( \frac { 2 } { 3 } \right) ^ { i + 1 }$$

Accepted Answer

A)  $\frac { 76 } { 81 }$ 
B)  $\frac { 38 } { 27 }$ 
C)  $\frac { 80 } { 81 }$ 
D)  $\frac { 520 } { 81 }$

Question 13

Expand a Binomial Raised to a Power
-A company models its yearly expenses in millions of dollars using the equation $f ( t ) = 0.07 t ^ { 3 } - 0.8 t ^ { 2 } + 1.21 t + 3.3$ where $t = 0$ represents 1988 . The company's account manager decides to adjust the model so that $t = 0$ corresponds to 1998 rather than 1988 . To do this, she obtains $g ( t ) = f ( t + 10 )$. Use the Binomial Theorem to express $g ( t )$ in descending powers of $t$.

Accepted Answer

A)  $0.07 \mathrm { t } ^ { 3 } + 1.3 \mathrm { t } ^ { 2 } + 6.21 \mathrm { t } + 5.4$ 
B)  $0.07 t ^ { 3 } - 2.9 t ^ { 2 } + 38.21 t + 3.51$ 
C)  $0.07 \mathrm { t } ^ { 3 } + 1.3 \mathrm { t } ^ { 2 } + 38.21 \mathrm { t } + 8.1$ 
D)  $0.07 t ^ { 3 } - 2.9 t ^ { 2 } - 0.79 t + 24.1$

Question 14

Use the Formula for the Sum of the First n Terms of an Arithmetic Sequence
-Find the sum of the first 40 terms of the arithmetic sequence: -20, -10, 0, 10, . . .

Accepted Answer

A)  7000 
B)  7006 
C)  380 
D)  7200

Question 15

Use Factorial Notation
-$$a _ { n } = 3 ( n + 2 ) !$$

Accepted Answer

A)  $18,72,360,2160$ 
B)  $18,144,1080,8640$ 
C)  $6,36,216,1440$ 
D)  $6,18,72,360$

Question 16

Find the Probability of One Event or a Second Event Occurring
-Each of ten tickets is marked with a different number from 1 to 10 and put in a box. If you draw a ticket from the box, what is the probability that you will draw 5, 6, or 1 ?

Accepted Answer

A)  $\frac { 3 } { 10 }$ 
B)  $\frac { 1 } { 5 }$ 
C)  $\frac { 1 } { 10 }$ 
D)  $\frac { 1 } { 6 }$

Question 17

Expand a Binomial Raised to a Power
-$$( 3 x - 1 ) ^ { 5 }$$

Accepted Answer

A)  $243 x ^ { 5 } - 405 x ^ { 4 } + 270 x ^ { 3 } - 90 x ^ { 2 } + 15 x - 1$ 
B)  $243 x ^ { 5 } - 81 x ^ { 4 } + 27 x ^ { 3 } - 9 x ^ { 2 } + 3 x - 1$ 
C)  $\left( 9 x ^ { 2 } - 6 x + 1 \right) ^ { 5 }$ 
D)  $243 x ^ { 5 } + 15 x ^ { 4 } - 90 x ^ { 3 } - 90 x ^ { 2 } + 15 x - 1$

Question 18

Use the Formula for the General Term of an Arithmetic Sequence
Choose the one alternative that best completes the statement or answers the question.
Use the formula for the general term (the nth term) of an arithmetic sequence to find the indicated term of the
sequence with the given first term, a1, and common difference, d.
-Find a8 when $a _ { 1 } = - 10 , d = - 4$.

Accepted Answer

A)  $- 38$ 
B)  $- 42$ 
C)  18 
D)  22

Question 19

A statement Sn about the positive integers is given. Write statements S1, S2, and S3, and show that each of these
statements is true.
-$$S _ { n } : 1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \ldots + n ( n + 1 ) = \frac { n ( n + 1 ) ( n + 2 ) } { 3 }$$

Accepted Answer

The answer of A statement Sn about the positive integers...

Question 20

Use the Formula for the General Term of a Geometric Sequence
-Find $a _ { 12 }$ when $a _ { 1 } = - 3 , r = - 2$.

Accepted Answer

A)  6144 
B)  $- 12,288$ 
C)  6148 
D)  $- 25$

Exam 11: Sequences, Induction, and Probability

Use Recursion Formulas - $a _ { 1 } = - 6$ and $a _ { n } = - 4 a _ { n - 1 }$ for $n \geq 2$

The general term of a sequence is given. Determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. - $a _ { n } = 5 ^ { n }$

Additional Concepts - $\frac { { } _ { 7 } C _ { 3 } } { { } _ { 5 } C _ { 1 } } - \frac { 48 ! } { 46 ! }$

Write Terms of an Arithmetic Sequence - $a _ { 1 } = - \frac { 3 } { 8 } ; d = - \frac { 1 } { 8 }$

Write the word or phrase that best completes each statement or answers the question. Use mathematical induction to prove that the statement is true for every positive integer n. - $8 + 16 + 24 + \ldots + 8 n = 4 n ( n + 1 )$

Find the Probability that an Event Will Not Occur -A bag contains 3 blue marbles, 8 green marbles, and 5 red marbles. One marble is drawn from the bag. What is the probability that the marble drawn is not blue?

Probability 1 Compute Empirical Probability -During July in Jacksonville, Florida, it is not uncommon to have afternoon thunderstorms. On average, 12.3 days have afternoon thunderstorms. What is the probability that a randomly selected day in July will not have a thunderstorm?

Use the Formula for the General Term of a Geometric Sequence -Find a 8 when $a _ { 1 } = 8,000,000 , r = 0.1$ .

Find the Probability of One Event and a Second Event Occurring -A 6 -sided die is rolled. What is the probability of rolling a number that is even and a 5 ?

Express the sum using summation notation. Use a lower limit of summation not necessarily 1 and k for the index of summation. - $4 + \frac { 9 } { 2 } + 5 + \frac { 11 } { 2 } + \ldots + 9$

Use the Permutations Formula -The matching section of an exam has 5 questions and 7 possible answers. In how many different ways can a student answer the 5 questions, if none of the answer choices can be repeated?

Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence. - $\sum _ { i = 1 } ^ { 3 } \left( \frac { 2 } { 3 } \right) ^ { i + 1 }$

Use the Formula for the Sum of the First n Terms of an Arithmetic Sequence -Find the sum of the first 40 terms of the arithmetic sequence: -20, -10, 0, 10, . . .

Use Factorial Notation - $a _ { n } = 3 ( n + 2 ) !$

Find the Probability of One Event or a Second Event Occurring -Each of ten tickets is marked with a different number from 1 to 10 and put in a box. If you draw a ticket from the box, what is the probability that you will draw 5, 6, or 1 ?

Expand a Binomial Raised to a Power - $( 3 x - 1 ) ^ { 5 }$

A statement Sn about the positive integers is given. Write statements S1, S2, and S3, and show that each of these statements is true. - $S _ { n } : 1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \ldots + n ( n + 1 ) = \frac { n ( n + 1 ) ( n + 2 ) } { 3 }$

Use the Formula for the General Term of a Geometric Sequence -Find $a _ { 12 }$ when $a _ { 1 } = - 3 , r = - 2$ .

Exam 11: Sequences, Induction, and Probability

Use Recursion Formulas - a1=−6a _ { 1 } = - 6a1​=−6 and an=−4an−1a _ { n } = - 4 a _ { n - 1 }an​=−4an−1​ for n≥2n \geq 2n≥2

The general term of a sequence is given. Determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. - an=5na _ { n } = 5 ^ { n }an​=5n

Additional Concepts - 7C35C1−48!46!\frac { { } _ { 7 } C _ { 3 } } { { } _ { 5 } C _ { 1 } } - \frac { 48 ! } { 46 ! }5​C1​7​C3​​−46!48!​

Write Terms of an Arithmetic Sequence - a1=−38;d=−18a _ { 1 } = - \frac { 3 } { 8 } ; d = - \frac { 1 } { 8 }a1​=−83​;d=−81​

Write the word or phrase that best completes each statement or answers the question. Use mathematical induction to prove that the statement is true for every positive integer n. - 8+16+24+…+8n=4n(n+1)8 + 16 + 24 + \ldots + 8 n = 4 n ( n + 1 )8+16+24+…+8n=4n(n+1)