Question 1

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.
-$$a _ { n } = 4 n ^ { 2 } - 3$$

Accepted Answer

The given formula $a_n = 4n^2 - 3$ does not represent a geometric sequence because the nth term does not take the form of $a_1 \cdot r^{(n-1)}$, where $a_1$ is the first term and $r$ is the common ratio. Instead, it represents a quadratic sequence, which is neither arithmetic (where each term is obtained by adding a constant difference to the previous term) nor geometric (where each term is obtained by multiplying the previous term by a constant ratio).

Question 2

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.
-$$a _ { n } = 5 n ^ { 2 }$$

Accepted Answer

The given sequence $a_n = 5n^2$ does not fit the form of a geometric sequence, which is $a_n = a_1 \cdot r^{n-1}$, because the term $5n^2$ suggests that each term is proportional to the square of its position, not a constant ratio times the previous term.

Question 3

Solve the problem.
-A pendulum bob swings through an arc 80 inches long on its first swing. Each swing thereafter, it swings only 86% as far as on the previous swing. What is the length of the arc after 11 swings? Round to two
Decimal places.

Accepted Answer

The length of the arc after 11 swings is 17.70 inches.The length of the arc after each swing is 86% of the length of the arc on the previous swing. So, the length of the arc after 11 swings is 0.86^11 * 80 = 17.70 inches.

Question 4

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

The sum of the first n terms of a geometric sequence can be found using the formula $S_n = \frac{a(1 - r^n)}{1 - r}$, where $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms. For this sequence, $a = 4$, $r = -3$ (since each term is -3 times the previous one), and $n = 11$. Plugging these values into the formula gives $S_{11} = \frac{4(1 - (-3)^{11})}{1 - (-3)} = \frac{4(1 + 177147)}{4} = 177148$.

Question 5

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.
-$$- \frac { 1 } { 6 } , \frac { 1 } { 54 } , - \frac { 1 } { 486 } , \frac { 1 } { 4374 } , \ldots$$

Accepted Answer

The formula for the nth term of a geometric sequence is $a_n = a_1 \cdot r^{n-1}$, where $a_1$ is the first term and $r$ is the common ratio. Here, $a_1 = -\frac{1}{6}$ and the common ratio $r$ can be found by dividing the second term by the first term, which gives $r = \frac{1/54}{-1/6} = -\frac{1}{9}$. Thus, the correct formula is $a_n = -\frac{1}{6} \cdot (-\frac{1}{9})^{n-1}$.

Question 6

Solve the problem.
-Carla 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 Carla's salary, to the nearest dollar, in the sixth year?

Accepted Answer

Carla's salary increases by 4% annually starting in the second year. To find her salary in the sixth year, we use the formula for compound interest: $A = P(1 + r)^n$, where $A$ is the amount after $n$ years, $P$ is the principal amount (initial salary), $r$ is the annual rate of increase, and $n$ is the number of years the salary has been increasing. Here, $P = $40,000$, $r = 0.04$ (4%), and $n = 5$ (since the salary increases start in the second year). Plugging these values in, we get $A = 40000(1 + 0.04)^5 = 40000(1.04)^5 ≈ $48,666$, rounding to the nearest dollar.

Question 7

Use the Formula for the Sum of the First n Terms of a Geometric Sequence
-Find the sum of the first five terms of the geometric sequence: $\frac { 3 } { 3 } , \frac { 6 } { 3 } , \frac { 12 } { 3 } , \ldots$

Accepted Answer

The answer of Use the Formula for the Sum of...

Question 8

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.
-$$a _ { n } = \left( \frac { 3 } { 2 } \right) ^ { n }$$

Accepted Answer

The given formula for the nth term of the sequence is of the form $$a_n = a_1 \times r^{n-1}$$, which is the formula for a geometric sequence. In this case, the first term $$a_1 = \frac{3}{2}$$ and the common ratio $$r = \frac{3}{2}$$. Therefore, the given sequence is a geometric sequence with common ratio $$r=\frac{3}{2}$$.

Question 9

Use the Formula for the Sum of the First n Terms of a Geometric Sequence
-Find the sum of the first 9 terms of the geometric sequence: 3, 9, 27, 81, 243, . . . .

Accepted Answer

The first term (a) is 3, the common ratio (r) is 3, and the number of terms (n) is 9. Using the formula for the sum of the first n terms of a geometric sequence, S_n = a(1-r^n)/(1-r), we get S_9 = 3(1-3^9)/(1-3) = 29,523.

Question 10

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

The first term is -4 and the common ratio is -3. So, the sum of the first 11 terms is:S11 = -4(1 - (-3)^11) / (1 - (-3))= -4(1 - (-3)^11) / 4= -4(1 - (-3)^11)= -4(1 - 177147)= -4(-177146)= 708584= -354,292

Question 11

Solve the problem.
-Ms. Patterson proposes to give her daughter Claire an allowance of $0.05 on the first day of her 13-day vacation, $0.10 on the second day, $0.20 on the third day, and so on. Find the allowance Claire would
Receive on the last day of her vacation.

Accepted Answer

The allowance doubles each day, starting at $0.05. On the 13th day, it would be $0.05 * 2^(13-1) = $0.05 * 2^12 = $0.05 * 4096 = $204.80.

Question 12

Solve the problem.
-The following table shows a country's population from 1995 to 1998:

Divide the population for each year by the population in the preceding year. Use this ratio to write the general term of the geometric sequence describing the country's population growth n years after $1994 .$ Then estimate the country's population, in millions, in 2004 .

Accepted Answer

The answer of Solve the problem.
-The following table shows a...

Question 13

Solve the problem.
-A football player signs a contract with a starting salary of $810,000 per year and an annual increase of 5.5% beginning in the second year. What will the athlete's salary be, to the nearest dollar, in the seventh year?

Accepted Answer

The salary in the seventh year is calculated using the formula for compound interest: $ S = P(1 + r)^n $, where $ P = 810,000 $, $ r = 0.055 $, and $ n = 6 $. This gives $ S = 810,000 \times (1.055)^6 \approx 1,116,863 $.

Question 14

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.
-$$a _ { n } = 5 n - 3$$

Accepted Answer

The formula for the general term of a geometric sequence is $a_n = a_1 \cdot r^{n-1}$, where $a_1$ is the first term and $r$ is the common ratio. In this case, $a_1 = 5$ and $r = 5$, so the general term is $a_n = 5 \cdot 5^{n-1} = 5n - 3$. Therefore, the sequence is geometric with common ratio $r = 5$.

Question 15

Use the Formula for the Sum of the First n Terms of a Geometric Sequence
-Find the sum of the first five terms of the geometric sequence: $\frac { 3 } { 3 } , \frac { 3 } { 6 } , \frac { 3 } { 12 } , \ldots .$

Accepted Answer

The sum of the first five terms of a geometric sequence can be found using the formula $S_n = \frac{a(1 - r^n)}{1 - r}$, where $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms. Here, $a = \frac{3}{3} = 1$, $r = \frac{3/6}{3/3} = \frac{1}{2}$, and $n = 5$. Plugging these values into the formula gives $S_5 = \frac{1(1 - (\frac{1}{2})^5)}{1 - \frac{1}{2}} = \frac{1 - \frac{1}{32}}{\frac{1}{2}} = \frac{31}{32} \times 2 = \frac{31}{16}$.

Question 16

Use the Formula for the Sum of the First n Terms of a Geometric Sequence
-Find the sum of the first five terms of the geometric sequence: -3, -6, -12, . . . .

Accepted Answer

The formula for the sum of the first n terms of a geometric sequence is $S_n = \frac{a(1 - r^n)}{1 - r}$, where $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms. Here, $a = -3$, $r = -6 / -3 = 2$, and $n = 5$. Plugging these values into the formula gives $S_5 = \frac{-3(1 - 2^5)}{1 - 2} = \frac{-3(1 - 32)}{-1} = \frac{-3(-31)}{-1} = -93$.

Question 17

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.
-$$\frac { 1 } { 5 } , - \frac { 1 } { 40 } , \frac { 1 } { 320 } , - \frac { 1 } { 2560 } , \ldots$$

Accepted Answer

The sequence is geometric with the first term $a_1 = \frac{1}{5}$ and a common ratio $r = -\frac{1}{8}$ (since each term is $-\frac{1}{8}$ times the previous term). The formula for the nth term of a geometric sequence is $a_n = a_1 \cdot r^{n-1}$, which matches choice A when substituting $a_1 = \frac{1}{5}$ and $r = -\frac{1}{8}$.

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.
-$$0.007,0.14,2.8,56 , \ldots$$

Accepted Answer

The given sequence is a geometric sequence because each term is obtained by multiplying the previous term by the same number (common ratio) of 20.

To find the nth term of a geometric sequence, we use the formula:

$$a_n=a_1r^{n-1}$$

where $$a_1$$ is the first term and r is the common ratio.

Using the formula,

$$a_1=0.007$$

$$r=20$$

Therefore,

$$a_n=0.007(20)^{n-1}$$

To find the 4th term, we substitute n=4 in the formula:

$$a_4=0.007(20)^{4-1}=0.007(20)^3=56$$

Hence, the answer is A.

Question 19

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.
-$$a _ { n } = 3 ^ { n }$$

Accepted Answer

The formula for the general term of a geometric sequence is: $$a_n = a_1 \cdot r^{n-1}$$ 
Comparing this formula with the given sequence, we can see that:
$$a_1 = 3^{1} = 3$$ 
$$r = 3$$
So, the formula becomes:
$$a_n = 3 \cdot 3^{n-1} = 3^n$$
which matches the given sequence. Therefore, this is a geometric sequence with common ratio r = 3.

Question 20

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

The formula for the sum of the first n terms of a geometric sequence is $S_n = \frac{a(1 - r^n)}{1 - r}$, where $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms. Here, $a = 4$, $r = 2$, and $n = 6$. Plugging these values into the formula gives $S_6 = \frac{4(1 - 2^6)}{1 - 2} = \frac{4(1 - 64)}{-1} = \frac{-4 \times 63}{-1} = 252$.

Quiz 8: Sequences, Induction, and Probability

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. - $a _ { n } = 4 n ^ { 2 } - 3$

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. - $a _ { n } = 5 n ^ { 2 }$

Solve the problem. -A pendulum bob swings through an arc 80 inches long on its first swing. Each swing thereafter, it swings only 86% as far as on the previous swing. What is the length of the arc after 11 swings? Round to two Decimal places.

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, . . . .

Solve the problem. -Carla 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 Carla's salary, to the nearest dollar, in the sixth year?

Use the Formula for the Sum of the First n Terms of a Geometric Sequence -Find the sum of the first five terms of the geometric sequence: $\frac { 3 } { 3 } , \frac { 6 } { 3 } , \frac { 12 } { 3 } , \ldots$

Use the Formula for the Sum of the First n Terms of a Geometric Sequence -Find the sum of the first 9 terms of the geometric sequence: 3, 9, 27, 81, 243, . . . .

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, . . . .

Solve the problem. -Ms. Patterson proposes to give her daughter Claire an allowance of $0.05 on the first day of her 13-day vacation, $0.10 on the second day, $0.20 on the third day, and so on. Find the allowance Claire would Receive on the last day of her vacation.

Solve the problem. -A football player signs a contract with a starting salary of $810,000 per year and an annual increase of 5.5% beginning in the second year. What will the athlete's salary be, to the nearest dollar, in the seventh year?

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. - $a _ { n } = 5 n - 3$

Use the Formula for the Sum of the First n Terms of a Geometric Sequence -Find the sum of the first five terms of the geometric sequence: $\frac { 3 } { 3 } , \frac { 3 } { 6 } , \frac { 3 } { 12 } , \ldots .$

Use the Formula for the Sum of the First n Terms of a Geometric Sequence -Find the sum of the first five terms of the geometric sequence: -3, -6, -12, . . . .

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. - $0.007,0.14,2.8,56 , \ldots$

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. - $a _ { n } = 3 ^ { n }$

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, . . . .