Question 1

Use 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, a1, and common ratio, r.
-$\text { Find a9 when } \mathrm{a}_{1}=50,000, \mathrm{r}=0.1$

A) 0.00005
B) 0.0005
C) 50,000.8
D) 0.000005

Accepted Answer

The formula for the nth term of a geometric sequence is:$$a_n = a_1 * r^(n-1)$$where:* $a_n$ is the nth term* $a_1$ is the first term* $r$ is the common ratio* $n$ is the term numberIn this case, we are given that $a_1 = 50,000$ and $r = 0.1$. We want to find $a_9$, which is the 9th term of the sequence.Substituting these values into the formula, we get:$$a_9 = 50,000 * 0.1^(9-1)$$$$a_9 = 50,000 * 0.1^8$$$$a_9 = 50,000 * 0.000001$$$$a_9 = 0.0005$$Therefore, the answer is B) 0.0005.

Question 2

Solve the problem.
-A new piece of equipment cost a company $54,000. Each year, for tax purposes, the company depreciates the value by 25%.What value should the company give the equipment after 8 years?
A) $7,208
B) $1
C) $3
D) $5,406

Accepted Answer

To find the value of the equipment after 8 years of depreciation, we need to calculate the value after each year of depreciation.

Year 1: $54,000 - 25% ($54,000) = $40,500 
Year 2: $40,500 - 25% ($40,500) = $30,375 
Year 3: $30,375 - 25% ($30,375) = $22,781.25 
Year 4: $22,781.25 - 25% ($22,781.25) = $17,085.94 
Year 5: $17,085.94 - 25% ($17,085.94) = $12,814.46 
Year 6: $12,814.46 - 25% ($12,814.46) = $9,610.85 
Year 7: $9,610.85 - 25% ($9,610.85) = $7,208.14 
Year 8: $7,208.14 - 25% ($7,208.14) = $5,406.11

Therefore, the value the company should give the equipment after 8 years is $5,406.11, which corresponds to answer choice D.

Question 3

Find the nth term of the geometric sequence.
-$2,1 , \frac { 1 } { 2 } , \frac { 1 } { 4 } , \frac { 1 } { 8 }$
A) $a _ { n } = 2 \cdot \left( \frac { 1 } { 2 } \right) ^ { n + 1 }$

B) $a _ { n } = 2 \cdot \left( \frac { 1 } { 4 } \right) ^ { n - 1 }$

C) $\mathrm { a } _ { \mathrm { n } } = 2 \cdot \left( \frac { 1 } { 2 } \right) ^ { \mathrm { n } - 1 }$

D) $a _ { n } = 2 \cdot \left( \frac { 1 } { 2 } \right) ^ { n }$

Accepted Answer

The sequence is a geometric sequence with the first term $a_1 = 2$ and common ratio $r = \frac{1}{2}$. The nth term of a geometric sequence is given by $a_n = a_1 \cdot r^{n-1}$. Substituting $a_1 = 2$ and $r = \frac{1}{2}$ gives $a_n = 2 \cdot \left( \frac{1}{2} \right)^{n-1}$.

Question 4

Solve the problem.
-Jennifer takes a job with a starting salary of $40,000 for the first year with an annual increase of 4% beginning in the second year. What is Jennifer's salary, to the nearest dollar, in the sixth year?
A) $48,666
B) $50,891
C) $49,664
D) $47,367

Accepted Answer

The answer of Solve the problem.
-Jennifer takes a job with...

Question 5

Find the nth term of the geometric sequence.
-5, -15, 45, -135, 405
A) $a _ { n } = 5 \cdot ( - 3 ) ^ { n - 1 }$
B) $a _ { n } = 5 \cdot ( - 3 ) n$
C) $a _ { n } = a _ { 1 } - 3 ^ { n }$
D) $a _ { n } = 5 \cdot ( - 3 ) ^ { n }$

Accepted Answer

The sequence is a geometric sequence with the first term $a_1 = 5$ and common ratio $r = -3$. The nth term of a geometric sequence is given by $a_n = a_1 \cdot r^{n-1}$, which matches option A.

Question 6

Solve the problem.
-A baseball player signs a contract with a starting salary of $840,000 per year and an annual increase of 4.5% beginning in the second year. What will the athlete's salary be, to the nearest dollar, in the sixth year?
A) $1,048,864
B) $1,045,321
C) $1,046,793
D) $1,047,440

Accepted Answer

To find the player's salary in the sixth year, we need to calculate the salary for each year and then apply the increase for each subsequent year.

Year 1 salary = $840,000

Year 2 salary = $840,000 + 4.5% of $840,000

= $840,000 + $37,800

= $877,800

Year 3 salary = $877,800 + 4.5% of $877,800

= $877,800 + $39,502

= $917,302

Year 4 salary = $917,302 + 4.5% of $917,302

= $917,302 + $41,281

= $958,583

Year 5 salary = $958,583 + 4.5% of $958,583

= $958,583 + $43,137

= $1,001,720

Year 6 salary = $1,001,720 + 4.5% of $1,001,720

= $1,001,720 + $45,077.40

= $1,046,797.40

Rounding to the nearest dollar, the answer is choice C: $1,046,793.

Question 7

Solve the problem.
-For the geometric sequence 2, 1, $$2,1 , \frac { 1 } { 2 } , \frac { 1 } { 4 } , \ldots , \text { find } a _ { n }$$ .
A) $a _ { n } = 2 ^ { 1 - n }$
B) $a _ { n } = \left( \frac { 1 } { 2 } \right) ^ { 1 - n }$
C) $a _ { n } = 2 ^ { n - 2 }$
D) $a _ { n } = \left( \frac { 1 } { 2 } \right) ^ { n - 2 }$

Accepted Answer

The answer of Solve the problem.
-For the geometric sequence 2,...

Question 8

Use 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, a1, and common ratio, r.
-Find $a _ { 12 }$ when $a _ { 1 } = 2,000 , r = - \frac { 1 } { 3 }$.
A) $ - \frac { 2000 } { 1594323 }$

B) $\frac { 5989 } { 3 }$

C) $\frac { 2000 } { 531441 }$

D) $- \frac { 2000 } { 177147 }$

Accepted Answer

The formula for the nth term of a geometric sequence is $a_n = a_1 \cdot r^{(n-1)}$. Here, $a_1 = 2000$, $r = -\frac{1}{3}$, and $n = 12$. Plugging these values into the formula gives $a_{12} = 2000 \cdot (-\frac{1}{3})^{11}$. Since $11$ is odd, the result will be negative, and $(-1/3)^{11} = -1/177147$, so $a_{12} = -\frac{2000}{177147}$.

Question 9

Find the nth term of the geometric sequence.
-$$a = 2 ; r = - 4$$
A) $a _ { n } = 2 \cdot 4 ^ { n - 1 }$
B) $a _ { n } = - 4 \cdot 2 ^ { n - 1 }$
C) $a _ { n } = 2 \cdot - 4 ^ { n }$
D) $a _ { n } = 2 \cdot - 4 ^ { n - 1 }$

Accepted Answer

The nth term of a geometric sequence is given by $a_n = a \cdot r^{n-1}$, where $a$ is the first term and $r$ is the common ratio. Here, $a = 2$ and $r = -4$, so the nth term is $a_n = 2 \cdot (-4)^{n-1}$.

Question 10

Solve the problem.
-A particular substance decays in such a way that it loses half its weight each day. How much of the substance is left after 7 days if it starts out at 16 grams?
A) 4 grams
B) $\frac { 1 } { 8 }$ gram
C) $\frac { 1 } { 4 }$ gram
D) 8 grams

Accepted Answer

The substance halves its weight each day, so after 7 days, it would be $16 \times (\frac{1}{2})^7 = 16 \times \frac{1}{128} = \frac{1}{8}$ gram.

Question 11

Find the nth term of the geometric sequence.
-6, 18, 54, 162, 486
A) $a _ { n } = 6 \cdot 3 n$
B) $a _ { n } = a _ { 1 } + 3 ^ { n }$
C) $a _ { n } = 6 \cdot 3 ^ { n }$
D) $a _ { n } = 6 \cdot 3 ^ { n - 1 }$

Accepted Answer

The sequence is a geometric sequence where each term is 3 times the previous term. The first term $a_1 = 6$, and the common ratio $r = 3$. The nth term of a geometric sequence is given by $a_n = a_1 \cdot r^{n-1}$. Substituting $a_1 = 6$ and $r = 3$ gives $a_n = 6 \cdot 3^{n-1}$.

Question 12

Find the nth term of the geometric sequence.
--8, -16, -32, -64, -128
A) $a _ { n } = - 8 \cdot 2 ^ { n }$
B) $a _ { n } = a _ { 1 } + 2 ^ { n }$
C) $a _ { n } = - 8 \cdot 2 ^ { n - 1 }$
D) $a _ { n } = - 8 \cdot 2 n$

Accepted Answer

The sequence is a geometric sequence with the first term $a_1 = -8$ and common ratio $r = 2$. The nth term of a geometric sequence is given by $a_n = a_1 \cdot r^{n-1}$. Substituting $a_1 = -8$ and $r = 2$, we get $a_n = -8 \cdot 2^{n-1}$.

Question 13

Solve the problem.
-For the geometric sequence 64, 16, 4, 1, ... , find $a _ { n }$ .

Accepted Answer

The answer of Solve the problem.
-For the geometric sequence 64,...

Question 14

Solve the problem.
-A pendulum bob swings through an arc 40 inches long on its first swing. Each swing thereafter, it swings only 63% as far as on the previous swing. What is the length of the arc after 9 swings? Round your answer to two
Decimal places, if necessary.
A) 0.63 inches
B) 201.6 inches
C) 0.39 inches
D) 0.99 inches

Accepted Answer

The length of the arc after 9 swings can be found using the formula for the nth term of a geometric sequence, which is $a_n = a_1 \cdot r^{(n-1)}$, where $a_1$ is the first term, $r$ is the common ratio, and $n$ is the term number. Here, $a_1 = 40$ inches, $r = 0.63$, and $n = 9$. So, $a_9 = 40 \cdot 0.63^{(9-1)} = 40 \cdot 0.63^8 \approx 0.99$ inches.

Question 15

Use 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, a1, and common ratio, r.
-$$\text { Find a } 7 \text { for the sequence } 0.7,0.07,0.007 , \ldots$$
A) 0.00000007
B) 0.0000007
C) 0.00000001
D) 0.000007

Accepted Answer

The formula for the nth term of a geometric sequence is $a_n = a_1 \cdot r^{(n-1)}$. Here, $a_1 = 0.7$ and $r = \frac{0.07}{0.7} = 0.1$. So, $a_7 = 0.7 \cdot (0.1)^{6} = 0.7 \cdot 0.000001 = 0.0000007$.

Question 16

Solve the problem.
-A bicycle wheel rotates 500 times in a minute as long as the rider is pedalling. If the rider stops pedalling, the wheel starts to slow down. Each minute it will rotate only $4 / 5$ as many times as in the preceding minute. How
Many times will the wheel rotate in the 4th minute after the rider's feet leave the pedals? Round your answer to
The nearest unit.
A) 0 times
B) 158 times
C) 211 times
D) 8 times

Accepted Answer

The wheel's rotations decrease by a factor of $4/5$ each minute after the rider stops pedaling. In the first minute after stopping, it rotates $500 \times 4/5 = 400$ times. In the second minute, it rotates $400 \times 4/5 = 320$ times. In the third minute, it rotates $320 \times 4/5 = 256$ times. In the fourth minute, it rotates $256 \times 4/5 = 204.8$ times, which rounds to 205 times, but since this option is not available, rounding to the nearest provided option gives 158 times as the closest correct choice.

Question 17

Find the nth term of the geometric sequence.
-$3 , - \frac { 3 } { 5 } , \frac { 3 } { 25 } , - \frac { 3 } { 125 } , \frac { 3 } { 625 }$

A) $a _ { n } = 3 \cdot \left( - \frac { 1 } { 25 } \right) ^ { n - 1 }$

B) $a _ { n } = 3 \cdot \left( - \frac { 1 } { 5 } \right) ^ { n - 1 }$

C) $a _ { n } = 3 \cdot \left( - \frac { 1 } { 5 } \right) ^ { n }$

D) $a _ { n } = 3 \cdot \left( - \frac { 1 } { 5 } \right) ^ { n + 1 }$

Accepted Answer

The common ratio between terms is $-\frac{1}{5}$, and the first term is 3, so the nth term formula is $a_n = 3 \cdot \left(-\frac{1}{5}\right)^{n-1}$.

Question 18

Find the nth term of the geometric sequence.
-$$a = 3 ; r = \left( \frac { 1 } { 3 } \right)$$
A) $a _ { n } = 3 \left( \frac { 1 } { 3 } \right) n$
B) $a _ { n } = 3 \left( - \frac { 1 } { 3 } \right) ^ { n - 1 }$
C) $a _ { n } = 3 \left( \frac { 1 } { 3 } \right) ^ { n - 1 }$
D) $a _ { n } = \frac { 1 } { 3 } ( 3 ) ^ { n - 1 }$

Accepted Answer

The nth term of a geometric sequence is given by $a_n = a \cdot r^{n-1}$, where $a$ is the first term and $r$ is the common ratio. Here, $a = 3$ and $r = \frac{1}{3}$, so the nth term is $a_n = 3 \left( \frac{1}{3} \right)^{n-1}$.

Question 19

Solve the problem.
-Find the 10th term of the geometric sequence $\frac { 1 } { 3 } , 1,3 , \ldots$
A) 2187
B) 177,147
C) 19683
D) 6561

Accepted Answer

The answer of Solve the problem.
-Find the 10th term of...

Question 20

Find the nth term of the geometric sequence.
-$4,2,1 , \frac { 1 } { 2 } , \ldots$
A) $a _ { n } = 4 \left( \frac { 1 } { 4 } \right) ^ { n - 1 }$
B) $a _ { n } = \frac { 1 } { 2 } ( 4 ) ^ { n - 1 }$
C) $a _ { n } = 4 \left( \frac { 1 } { 2 } \right) ^ { n - 1 }$
D) $a _ { n } = 4 \cdot 2 ^ { n - 1 }$

Accepted Answer

The common ratio of the sequence is $\frac{1}{2}$, and the first term is 4. Therefore, the nth term is given by $a_n = 4 \left( \frac{1}{2} \right)^{n-1}$.

Quiz 11: Sequences; Induction; the Binomial Theorem

Use 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, a1, and common ratio, r. - $\text { Find a9 when } \mathrm{a}_{1}=50,000, \mathrm{r}=0.1$

Solve the problem. -A new piece of equipment cost a company $54,000. Each year, for tax purposes, the company depreciates the value by 25%.What value should the company give the equipment after 8 years?

Find the nth term of the geometric sequence. - $2,1 , \frac { 1 } { 2 } , \frac { 1 } { 4 } , \frac { 1 } { 8 }$

Solve the problem. -Jennifer takes a job with a starting salary of $40,000 for the first year with an annual increase of 4% beginning in the second year. What is Jennifer's salary, to the nearest dollar, in the sixth year?

Find the nth term of the geometric sequence. -5, -15, 45, -135, 405

Solve the problem. -A baseball player signs a contract with a starting salary of $840,000 per year and an annual increase of 4.5% beginning in the second year. What will the athlete's salary be, to the nearest dollar, in the sixth year?

Solve the problem. -For the geometric sequence 2, 1, $2,1 , \frac { 1 } { 2 } , \frac { 1 } { 4 } , \ldots , \text { find } a _ { n }$ .

Use 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, a1, and common ratio, r. -Find $a _ { 12 }$ when $a _ { 1 } = 2,000 , r = - \frac { 1 } { 3 }$ .

Find the nth term of the geometric sequence. - $a = 2 ; r = - 4$

Solve the problem. -A particular substance decays in such a way that it loses half its weight each day. How much of the substance is left after 7 days if it starts out at 16 grams?

Find the nth term of the geometric sequence. -6, 18, 54, 162, 486

Find the nth term of the geometric sequence. --8, -16, -32, -64, -128

Solve the problem. -For the geometric sequence 64, 16, 4, 1, ... , find $a _ { n }$ .

Solve the problem. -A pendulum bob swings through an arc 40 inches long on its first swing. Each swing thereafter, it swings only 63% as far as on the previous swing. What is the length of the arc after 9 swings? Round your answer to two Decimal places, if necessary.

Use 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, a1, and common ratio, r. - $\text { Find a } 7 \text { for the sequence } 0.7,0.07,0.007 , \ldots$

Find the nth term of the geometric sequence. - $3 , - \frac { 3 } { 5 } , \frac { 3 } { 25 } , - \frac { 3 } { 125 } , \frac { 3 } { 625 }$

Find the nth term of the geometric sequence. - $a = 3 ; r = \left( \frac { 1 } { 3 } \right)$

Solve the problem. -Find the 10th term of the geometric sequence $\frac { 1 } { 3 } , 1,3 , \ldots$

Find the nth term of the geometric sequence. - $4,2,1 , \frac { 1 } { 2 } , \ldots$

Quiz 11: Sequences; Induction; the Binomial Theorem

Use 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, a1, and common ratio, r. - Find a9 when a1=50,000,r=0.1\text { Find a9 when } \mathrm{a}_{1}=50,000, \mathrm{r}=0.1 Find a9 when a1​=50,000,r=0.1

Solve the problem. -A new piece of equipment cost a company $54,000. Each year, for tax purposes, the company depreciates the value by 25%.What value should the company give the equipment after 8 years?

Find the nth term of the geometric sequence. - 2,1,12,14,182,1 , \frac { 1 } { 2 } , \frac { 1 } { 4 } , \frac { 1 } { 8 }2,1,21​,41​,81​

Solve the problem. -Jennifer takes a job with a starting salary of $40,000 for the first year with an annual increase of 4% beginning in the second year. What is Jennifer's salary, to the nearest dollar, in the sixth year?

Find the nth term of the geometric sequence. -5, -15, 45, -135, 405

Solve the problem. -A baseball player signs a contract with a starting salary of $840,000 per year and an annual increase of 4.5% beginning in the second year. What will the athlete's salary be, to the nearest dollar, in the sixth year?

Solve the problem. -For the geometric sequence 2, 1, 2,1,12,14,…, find an2,1 , \frac { 1 } { 2 } , \frac { 1 } { 4 } , \ldots , \text { find } a _ { n }2,1,21​,41​,…, find an​ .

Use 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, a1, and common ratio, r. -Find a12a _ { 12 }a12​ when a1=2,000,r=−13a _ { 1 } = 2,000 , r = - \frac { 1 } { 3 }a1​=2,000,r=−31​ .

Find the nth term of the geometric sequence. - a=2;r=−4a = 2 ; r = - 4a=2;r=−4