Question 1

Use the Formula for the Sum of the First n Terms of a Geometric Sequence
-Find the sum of the first six terms of the geometric sequence: 4, 8, 16, . . . .

Accepted Answer

A) 252 
B) 28 
C) 8190 
D) 63 
A) 252 
B) 28 
C) 8190 
D) 63 A

Question 2

Write a formula for the general term (the nth term)of the arithmetic sequence. Then use the formula for $r a _ { n }$ to find $a _ { 20 }$ , the 20th term of the sequence.
-$2,6,10,14,18 , \ldots$

Accepted Answer

A)  $a _ { n } = 4 n - 2 ; a _ { 20 } = 78$ 
B)  $a _ { n } = 2 n - 4 ; a _ { 20 } = 36$ 
C)  $a _ { n } = 4 n + 2 ; a _ { 20 } = 82$ 
D)  $a _ { n } = n + 4 ; a _ { 20 } = 24$ 
A)  $a _ { n } = 4 n - 2 ; a _ { 20 } = 78$ 
B)  $a _ { n } = 2 n - 4 ; a _ { 20 } = 36$ 
C)  $a _ { n } = 4 n + 2 ; a _ { 20 } = 82$ 
D)  $a _ { n } = n + 4 ; a _ { 20 } = 24$ A

Question 3

Use the Formula for the Sum of the First n Terms of a Geometric Sequence
-Find the sum of the first 11 terms of the geometric sequence: 4, -12, 36, -108, 324, . . . .

Accepted Answer

A) 177,148 
B) 177,146 
C) 177,155 
D) 177,142 
A) 177,148 
B) 177,146 
C) 177,155 
D) 177,142 A

Question 4

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 4, 7, or 3?

Accepted Answer

A)  $\frac { 3 } { 10 }$ 
B)  $\frac { 1 } { 4 }$ 
C)  $\frac { 1 } { 10 }$ 
D)  $\frac { 1 } { 7 }$ 
A)  $\frac { 3 } { 10 }$ 
B)  $\frac { 1 } { 4 }$ 
C)  $\frac { 1 } { 10 }$ 
D)  $\frac { 1 } { 7 }$

Question 5

Find the Probability of One Event and a Second Event Occurring
-A card is drawn from a well-shuffled deck of 52 cards. What is the probability that the card will have a value of 3 and be a face card?

Accepted Answer

A)  0 
B)  $\frac { 1 } { 13 }$ 
C)  $\frac { 4 } { 13 }$ 
D)  $\frac { 3 } { 13 }$ 
A)  0 
B)  $\frac { 1 } { 13 }$ 
C)  $\frac { 4 } { 13 }$ 
D)  $\frac { 3 } { 13 }$

Question 6

Solve the problem.
-A company models its yearly expenses in millions of dollars using the equation $f ( t ) = 0.05 t ^ { 3 } - 0.6 t ^ { 2 } + 1.25 t + 2.5$ where $t = 0$ represents 1986 . The company's account manager decides to adjust the model so that $t = 0$ corresponds to 1991 rather than 1986 . To do this, she obtains $g ( t ) = f ( t + 5 )$. Use the Binomial Theorem to express $g ( t )$ in descending powers of $t$.

Accepted Answer

A)  $0.05 \mathrm { t } ^ { 3 } + 0.15 \mathrm { t } ^ { 2 } - 1 \mathrm { t } + 0$ 
B)  $0.05 \mathrm { t } ^ { 3 } - 1.35 \mathrm { t } ^ { 2 } + 11 \mathrm { t } + 2$ 
C)  $0.05 t ^ { 3 } + 0.15 t ^ { 2 } + 11 t + 6$ 
D)  $0.05 t ^ { 3 } - 1.35 t ^ { 2 } - 2.25 t + 12$ 3 Find a Particular Term in a Binomial Expansion
Write the first three terms in the binomial expansion, expressing the result in simplified form. 
A)  $0.05 \mathrm { t } ^ { 3 } + 0.15 \mathrm { t } ^ { 2 } - 1 \mathrm { t } + 0$ 
B)  $0.05 \mathrm { t } ^ { 3 } - 1.35 \mathrm { t } ^ { 2 } + 11 \mathrm { t } + 2$ 
C)  $0.05 t ^ { 3 } + 0.15 t ^ { 2 } + 11 t + 6$ 
D)  $0.05 t ^ { 3 } - 1.35 t ^ { 2 } - 2.25 t + 12$ 3 Find a Particular Term in a Binomial Expansion
Write the first three terms in the binomial expansion, expressing the result in simplified form.

Question 7

Solve the problem.
-As part of her retirement savings plan, Patricia deposited $150 in a bank account during her first year in the workforce. During each subsequent year, she deposited $45 more than the previous year. Find how
Much she deposited during her twentieth year in the workforce. Find the total amount deposited in the
Twenty years.

Accepted Answer

A) $1005; $11,550 
B) $1050; $12,000 
C) $1005; $23,100 
D) $1050; $24,000 
A) $1005; $11,550 
B) $1050; $12,000 
C) $1005; $23,100 
D) $1050; $24,000

Question 8

Use the Formula for the Sum of the First n Terms of an Arithmetic Sequence
-Find the sum of the first 55 terms of the arithmetic sequence: 2, 4, 6, 8, . . .

Accepted Answer

A) 3080 
B) 3086 
C) 112 
D) 3135 
A) 3080 
B) 3086 
C) 112 
D) 3135

Question 9

Use Summation Notation
-$$a + 1 + \frac { a + 2 } { 2 } + \ldots + \frac { a + 5 } { 5 }$$

Accepted Answer

A)  $\sum _ { i = 1 } ^ { 5 } \frac { a + i } { i }$ 
B)  $\sum _ { i = 0 } ^ { 5 } \frac { a + i } { i }$ 
C)  $\sum _ { i = 0 } ^ { n } \frac { a + i } { i }$ 
D)  $\sum _ { i = 1 } ^ { n } \frac { a + i } { i }$ 
A)  $\sum _ { i = 1 } ^ { 5 } \frac { a + i } { i }$ 
B)  $\sum _ { i = 0 } ^ { 5 } \frac { a + i } { i }$ 
C)  $\sum _ { i = 0 } ^ { n } \frac { a + i } { i }$ 
D)  $\sum _ { i = 1 } ^ { n } \frac { a + i } { i }$

Question 10

Use mathematical induction to prove that the statement is true for every positive integer n.
-$$4 + \frac { 4 } { 5 } + \frac { 4 } { 25 } + \ldots + \frac { 4 } { 5 ^ { n - 1 } } = 5 \left( 1 - \frac { 1 } { 5 ^ { n } } \right)$$

Accepted Answer

\[\begin{aligned}
S _ { 1 } : 4 & = 5 \l

Question 11

Use the Formula for the Sum of an Infinite Geometric Series
-$$2 - \frac { 1 } { 2 } + \frac { 1 } { 8 } - \frac { 1 } { 32 } + \ldots$$

Accepted Answer

A)  $\frac { 8 } { 5 }$ 
B)  $\frac { 3 } { 2 }$ 
C)  $- \frac { 1 } { 2 }$ 
D)  does not exist 
A)  $\frac { 8 } { 5 }$ 
B)  $\frac { 3 } { 2 }$ 
C)  $- \frac { 1 } { 2 }$ 
D)  does not exist

Question 12

Compute Empirical Probability
Solve the problem. Round to the nearest hundredth of a percent if needed.
-A traffic engineer is counting the number of vehicles by type that turn into a residential area. The table below shows the results of the counts during a four-hour period. What is the probability that the next
Vehicle passing is an SUV?

Accepted Answer

A)  $33.13 \%$ 
B)  $19.36 \%$ 
C)  $34.76 \%$ 
D)  $33.72 \%$ 
A)  $33.13 \%$ 
B)  $19.36 \%$ 
C)  $34.76 \%$ 
D)  $33.72 \%$

Question 13

Use the Formula for the General Term of a Geometric SequenceUse the formula for the general term (the nth term)of a geometric sequence to find the indicated term of the
sequence with the given first term, $$\mathbf { a } _ { 1 }$$ , and common ratio, r.
-Find a 11 when $a _ { 1 } = 2 , r = 3$.

Accepted Answer

A)  118,098 
B)  354,294 
C)  118,102 
D)  32 
A)  118,098 
B)  354,294 
C)  118,102 
D)  32

Question 14

Solve the problem.
-A deposit of $11,000 is made in an account that earns 8% interest compounded quarterly. The balance in the account after n quarters is given by the sequence $$\mathrm { a } _ { \mathrm { n } } = 11,000 \left( 1 + \frac { 0.08 } { 4 } \right) ^ { \mathrm { n } } , \mathrm { n } = 1,2,3 , \ldots$$
Find the balance in the account after 4 years.

Accepted Answer

A)  $\$ 15,100.64$ 
B)  $\$ 5491.14$ 
C)  $\$ 11,906.75$ 
D)  $\$ 4257.14$ 
A)  $\$ 15,100.64$ 
B)  $\$ 5491.14$ 
C)  $\$ 11,906.75$ 
D)  $\$ 4257.14$

Question 15

Compute Theoretical Probability
-A 6-sided die is rolled. What is the probability of rolling a number less than 6?

Accepted Answer

A)  $\frac { 5 } { 6 }$ 
B)  $\frac { 5 } { 7 }$ 
C)  $\frac { 1 } { 6 }$ 
D)  $\frac { 1 } { 3 }$ 
A)  $\frac { 5 } { 6 }$ 
B)  $\frac { 5 } { 7 }$ 
C)  $\frac { 1 } { 6 }$ 
D)  $\frac { 1 } { 3 }$

Question 16

Additional Concepts
-$$\frac { 4 ^ { C _ { 2 } \cdot { } _ { 6 } C _ { 1 } } } { 16 C _ { 13 } }$$

Accepted Answer

A)  $\frac { 9 } { 140 }$ 
B)  $\frac { 9 } { 70 }$ 
C)  $\frac { 3 } { 140 }$ 
D)  $\frac { 27 } { 28 }$ 
A)  $\frac { 9 } { 140 }$ 
B)  $\frac { 9 } { 70 }$ 
C)  $\frac { 3 } { 140 }$ 
D)  $\frac { 27 } { 28 }$

Question 17

Find the Probability that an Event Will Not Occur
-A bag contains 20 marbles, of which 6 are blue and 10 are green. One marble is drawn from the bag. What is the probability that the marble drawn is not blue?

Accepted Answer

A)  $\frac { 7 } { 10 }$ 
B)  $\frac { 3 } { 10 }$ 
C)  $\frac { 3 } { 8 }$ 
D)  $\frac { 4 } { 5 }$ 
A)  $\frac { 7 } { 10 }$ 
B)  $\frac { 3 } { 10 }$ 
C)  $\frac { 3 } { 8 }$ 
D)  $\frac { 4 } { 5 }$

Question 18

Use the Formula for the General Term of a Geometric SequenceUse the formula for the general term (the nth term)of a geometric sequence to find the indicated term of the
sequence with the given first term, $$\mathbf { a } _ { 1 }$$ , and common ratio, r.
-$3 , \frac { 3 } { 2 } , \frac { 3 } { 4 } , \frac { 3 } { 8 } , \frac { 3 } { 16 } , \ldots$

Accepted Answer

A)  $a _ { n } = 3 \left( \frac { 1 } { 2 } \right) ^ { n - 1 }$ 
B)  $a _ { n } = 3 \left( \frac { 1 } { 2 } \right) ^n$ 
C)  $a _ { n } = 3 \left( \frac { 1 } { 2 } \right)^{ n + 1}$ 
D)  $a _ { n } = 3 \left( \frac { 1 } { 4 } \right)^{ n - 1}$ 
A)  $a _ { n } = 3 \left( \frac { 1 } { 2 } \right) ^ { n - 1 }$ 
B)  $a _ { n } = 3 \left( \frac { 1 } { 2 } \right) ^n$ 
C)  $a _ { n } = 3 \left( \frac { 1 } { 2 } \right)^{ n + 1}$ 
D)  $a _ { n } = 3 \left( \frac { 1 } { 4 } \right)^{ n - 1}$

Question 19

Use Summation Notation
-$$\sum _ { i = 4 } ^ { 7 } \frac { i ! } { ( i - 1 ) ! }$$

Accepted Answer

A)  22 
B)  14 
C)  $\frac { 7 } { 3 }$ 
D)  7 
A)  22 
B)  14 
C)  $\frac { 7 } { 3 }$ 
D)  7

Question 20

Use Recursion Formulas
-$a _ { 1 } = - 5$ and $a _ { n } = a _ { n - 1 } - 3$ for $n \geq 2$

Accepted Answer

A)  $- 5 , - 8 , - 11 , - 14$ 
B)  $5,2 , - 1 , - 4$ 
C)  $5,8,11,14$ 
D)  $- 5 , - 4 , - 1,2$ 
A)  $- 5 , - 8 , - 11 , - 14$ 
B)  $5,2 , - 1 , - 4$ 
C)  $5,8,11,14$ 
D)  $- 5 , - 4 , - 1,2$

Use the Formula for the Sum of the First n Terms of a Geometric Sequence -Find the sum of the first six terms of the geometric sequence: 4, 8, 16, . . . .

Write a formula for the general term (the nth term)of the arithmetic sequence. Then use the formula for $r a _ { n }$ to find $a _ { 20 }$ , the 20th term of the sequence. - $2,6,10,14,18 , \ldots$

Use the Formula for the Sum of the First n Terms of a Geometric Sequence -Find the sum of the first 11 terms of the geometric sequence: 4, -12, 36, -108, 324, . . . .

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 4, 7, or 3?

Find the Probability of One Event and a Second Event Occurring -A card is drawn from a well-shuffled deck of 52 cards. What is the probability that the card will have a value of 3 and be a face card?

Use the Formula for the Sum of the First n Terms of an Arithmetic Sequence -Find the sum of the first 55 terms of the arithmetic sequence: 2, 4, 6, 8, . . .

Use Summation Notation - $a + 1 + \frac { a + 2 } { 2 } + \ldots + \frac { a + 5 } { 5 }$

Use mathematical induction to prove that the statement is true for every positive integer n. - $4 + \frac { 4 } { 5 } + \frac { 4 } { 25 } + \ldots + \frac { 4 } { 5 ^ { n - 1 } } = 5 \left( 1 - \frac { 1 } { 5 ^ { n } } \right)$

Use the Formula for the Sum of an Infinite Geometric Series - $2 - \frac { 1 } { 2 } + \frac { 1 } { 8 } - \frac { 1 } { 32 } + \ldots$

Use the Formula for the General Term of a Geometric SequenceUse the formula for the general term (the nth term)of a geometric sequence to find the indicated term of the sequence with the given first term, $\mathbf { a } _ { 1 }$ , and common ratio, r. -Find a 11 when $a _ { 1 } = 2 , r = 3$ .

Compute Theoretical Probability -A 6-sided die is rolled. What is the probability of rolling a number less than 6?

Additional Concepts - $\frac { 4 ^ { C _ { 2 } \cdot { } _ { 6 } C _ { 1 } } } { 16 C _ { 13 } }$

Find the Probability that an Event Will Not Occur -A bag contains 20 marbles, of which 6 are blue and 10 are green. One marble is drawn from the bag. What is the probability that the marble drawn is not blue?

Use Summation Notation - $\sum _ { i = 4 } ^ { 7 } \frac { i ! } { ( i - 1 ) ! }$

Use Recursion Formulas - $a _ { 1 } = - 5$ and $a _ { n } = a _ { n - 1 } - 3$ for $n \geq 2$

Equations and Inequalities

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Exam 8: Sequences, Induction, and Probability