Question 1

Provide an appropriate response.
-Convert the decimal $.100100100 \ldots$ to a fraction by writing it as the infinite series $\frac { 100 } { 1000 } + \frac { 100 } { 1000 ^ { 2 } } + \frac { 100 } { 1000 ^ { 3 } } + \ldots$

Accepted Answer

This decimal is a repeating decimal, which can be represented as a geometric series with the first term $a = \frac{100}{1000}$ and the common ratio $r = \frac{1}{1000}$. The sum of an infinite geometric series is given by $S = \frac{a}{1 - r}$, which in this case is $S = \frac{\frac{100}{1000}}{1 - \frac{1}{1000}} = \frac{100}{999}$.

Question 2

Find the future value of the annuity.
-There are payments of $1000 at the end of each year for 15 years at 3% interest compounded annually

Accepted Answer

The future value of an annuity can be calculated using the formula FV = P * [(1 + r)^n - 1] / r, where P is the payment amount, r is the annual interest rate, and n is the number of periods. Plugging in the values (P = $1000, r = 0.03, n = 15), we get FV = $1000 * [(1 + 0.03)^15 - 1] / 0.03, which calculates to approximately $18,598.91.

Question 3

Evaluate the expression.
-$$\left( \begin{array} { l } 
3 \
3
\end{array} \right)$$

Accepted Answer

The expression $\left( \begin{array} { l } 3 \ 3 \end{array} \right)$ is a binomial coefficient, which calculates the number of ways to choose 3 items out of 3, without regard to order. This value is always 1, as there is exactly one way to choose all items from a set of the same size.

Question 4

Find the future value of the annuity.
-A sequence of yearly payments of $6000 is invested at the end of each of 11 years at an interest rate of 2.1%, compounded annually. What is the total amount of the annuity after 11 years?

Accepted Answer

The answer of Find the future value of the annuity.
-A...

Question 5

Provide an appropriate response.
-Convert the decimal $.8262626 \ldots$ to a fraction by writing it as the infinite series
$$\frac { 8 } { 10 } + \frac { 26 } { 10 \cdot 100 } + \frac { 26 } { 10 \cdot 100 ^ { 2 } } + \frac { 26 } { 10 \cdot 100 ^ { 3 } } + \ldots$$

Accepted Answer

The given series represents the decimal expansion of 0.8262626... infinitely, which directly matches choice D, indicating the decimal form of the fraction.

Question 6

Find the future value of the annuity.
-To save for retirement, you decide to deposit $2250 into an IRA at the end of each year for the next 40 years. If the interest rate is 2% per year compounded annually, find the value of the IRA,
Rounded to the nearest dollar, after 40 years.

Accepted Answer

The answer of Find the future value of the annuity.
-To...

Question 7

Provide an appropriate response.
-The sum of an infinite geometric series exists if:

Accepted Answer

The sum of an infinite geometric series exists and is finite if and only if the absolute value of the common ratio $r$ is less than 1, i.e., $| r | < 1$. This condition ensures that the terms of the series approach zero as the number of terms increases.

Question 8

Find the future value of the annuity.
-There are payments of $100 at the end of each year for 11 years at 3% interest compounded annually

Accepted Answer

The future value of an annuity is the sum of the present values of each payment. The present value of each payment is the payment divided by (1 + r)^n, where r is the interest rate and n is the number of years. In this case, the present value of each payment is $100 / (1 + 0.03)^n. The future value of the annuity is the sum of the present values of each payment, which is $100 / (1 + 0.03)^n + $100 / (1 + 0.03)^(n-1) + ... + $100 / (1 + 0.03). This can be simplified to $100 * (1 - (1 + 0.03)^-11) / 0.03, which is $1280.78.

Question 9

Provide an appropriate response.
-Consider the sequence defined by a $a _ { n } = - 80 n - 58$ . Is this sequence arithmetic, geometric, or neither?

Accepted Answer

The sequence is arithmetic because there is a constant difference between each term. The difference is -80, which is the coefficient of n.

Question 10

Provide an appropriate response.
-Convert the decimal $.868686 \ldots$ to a fraction by writing it as the infinite series
$$\frac { 86 } { 100 } + \frac { 86 } { 100 ^ { 2 } } + \frac { 86 } { 100 ^ { 3 } } + \ldots$$

Accepted Answer

This series is a geometric series with the first term $a = \frac{86}{100}$ and common ratio $r = \frac{1}{100}$. The sum of an infinite geometric series is given by $S = \frac{a}{1 - r}$. Substituting $a = \frac{86}{100}$ and $r = \frac{1}{100}$ gives $S = \frac{\frac{86}{100}}{1 - \frac{1}{100}} = \frac{86}{99}$, which matches option C.

Question 11

Provide an appropriate response.
-What is the common ratio in the geometric sequence $\left. a _ { n } = \log \left[ 6 ^ { \left( 5 ^ { n - 1 } \right.} \right) \right]$ ?

Accepted Answer

The answer of Provide an appropriate response.
-What is the common...

Question 12

Provide an appropriate response.
-For $\frac { 14 } { 13 } , x , \frac { 8 } { 11 }$ to be a geometric sequence, $x$ must be:

Accepted Answer

In a geometric sequence, the ratio between consecutive terms is constant. Thus, $\frac{x}{\frac{14}{13}} = \frac{\frac{8}{11}}{x}$. Solving for $x$, we find $x = \sqrt{\frac{112}{143}}$.

Question 13

Evaluate the expression.
-${ } _ { 9 } \mathrm { C } _ { 2 }$

Accepted Answer

${ } _ { 9 } \mathrm { C } _ { 2 }$ calculates the number of combinations of 9 items taken 2 at a time. The formula for combinations is ${ } _ { n } \mathrm { C } _ { r } = \frac{n!}{r!(n-r)!}$. Plugging in 9 for $n$ and 2 for $r$, we get $\frac{9!}{2!(9-2)!} = \frac{9 \times 8}{2} = 36$.

Question 14

Provide an appropriate response.
-Let $f ( x ) = 8 \left( 6 ^ { x } \right)$, and consider the sequence $f ( 1 ) , f ( 2 ) , f ( 3 ) , \ldots$ Is this a geometric sequence, and if so, what is the common ratio?

Accepted Answer

The sequence is geometric because each term is obtained by multiplying the previous term by a constant ratio. The common ratio is found by dividing any term by the previous term: $f(2)/f(1) = 8(6^2)/8(6^1) = 6$, indicating that the common ratio is 6.

Question 15

Evaluate the expression.
-$$\left( \begin{array} { c } 
129 \
3
\end{array} \right)$$

Accepted Answer

The answer of Evaluate the expression.
-\[\left( \begin{array} { c }...

Question 16

Evaluate the expression.
-$$\left( \begin{array} { c } 
10 \
3
\end{array} \right)$$

Accepted Answer

The answer of Evaluate the expression.
-\[\left( \begin{array} { c }...

Question 17

Evaluate the expression.
-$$\frac { 7 ! } { 5 ! 2 ! }$$

Accepted Answer

The expression simplifies to $\frac{7 \times 6 \times 5!}{5! \times 2 \times 1}$, where $5!$ cancels out, leaving $\frac{7 \times 6}{2} = 21$.

Question 18

Provide an appropriate response.
-For $7 , x , 5$ to be an arithmetic sequence, $x$ must be:

Accepted Answer

The answer of Provide an appropriate response.
-For \(7 , x...

Question 19

Evaluate the expression.
-$$\frac { 10 ! } { 5 ! 5 ! }$$

Accepted Answer

The expression simplifies to $\binom{10}{5}$, which calculates the number of ways to choose 5 items out of 10, equal to 252.

Question 20

Find the common ratio r for the given infinite geometric sequence.
-A small start-up consulting company hopes to have 7 times the clientele it had the previous year for the next 2 years. If the company just meets its targets each of the 2 years and it had 7 clients to
Begin with, how many clients does it have at the end of 2 years?

Accepted Answer

The answer of Find the common ratio r for the...

Quiz 12: Further Topics in Algebra

Provide an appropriate response. -Convert the decimal $.100100100 \ldots$ to a fraction by writing it as the infinite series $\frac { 100 } { 1000 } + \frac { 100 } { 1000 ^ { 2 } } + \frac { 100 } { 1000 ^ { 3 } } + \ldots$

Find the future value of the annuity. -There are payments of $1000 at the end of each year for 15 years at 3% interest compounded annually

Evaluate the expression. - $\left( \begin{array} { l } 3 \\3\end{array} \right)$

Find the future value of the annuity. -A sequence of yearly payments of $6000 is invested at the end of each of 11 years at an interest rate of 2.1%, compounded annually. What is the total amount of the annuity after 11 years?

Provide an appropriate response. -Convert the decimal $.8262626 \ldots$ to a fraction by writing it as the infinite series $\frac { 8 } { 10 } + \frac { 26 } { 10 \cdot 100 } + \frac { 26 } { 10 \cdot 100 ^ { 2 } } + \frac { 26 } { 10 \cdot 100 ^ { 3 } } + \ldots$

Find the future value of the annuity. -To save for retirement, you decide to deposit $2250 into an IRA at the end of each year for the next 40 years. If the interest rate is 2% per year compounded annually, find the value of the IRA, Rounded to the nearest dollar, after 40 years.

Provide an appropriate response. -The sum of an infinite geometric series exists if:

Find the future value of the annuity. -There are payments of $100 at the end of each year for 11 years at 3% interest compounded annually

Provide an appropriate response. -Consider the sequence defined by a $a _ { n } = - 80 n - 58$ . Is this sequence arithmetic, geometric, or neither?

Provide an appropriate response. -Convert the decimal $.868686 \ldots$ to a fraction by writing it as the infinite series $\frac { 86 } { 100 } + \frac { 86 } { 100 ^ { 2 } } + \frac { 86 } { 100 ^ { 3 } } + \ldots$

Provide an appropriate response. -What is the common ratio in the geometric sequence $\left. a _ { n } = \log \left[ 6 ^ { \left( 5 ^ { n - 1 } \right.} \right) \right]$ ?

Provide an appropriate response. -For $\frac { 14 } { 13 } , x , \frac { 8 } { 11 }$ to be a geometric sequence, $x$ must be:

Evaluate the expression. - ${ } _ { 9 } \mathrm { C } _ { 2 }$

Provide an appropriate response. -Let $f ( x ) = 8 \left( 6 ^ { x } \right)$ , and consider the sequence $f ( 1 ) , f ( 2 ) , f ( 3 ) , \ldots$ Is this a geometric sequence, and if so, what is the common ratio?

Evaluate the expression. - $\left( \begin{array} { c } 129 \\3\end{array} \right)$

Evaluate the expression. - $\left( \begin{array} { c } 10 \\3\end{array} \right)$

Evaluate the expression. - $\frac { 7 ! } { 5 ! 2 ! }$

Provide an appropriate response. -For $7 , x , 5$ to be an arithmetic sequence, $x$ must be:

Evaluate the expression. - $\frac { 10 ! } { 5 ! 5 ! }$

Quiz 12: Further Topics in Algebra

Find the future value of the annuity. -There are payments of $1000 at the end of each year for 15 years at 3% interest compounded annually

Evaluate the expression. - (33) \left( \begin{array} { l } 3 \\3\end{array} \right) (33​)

Find the future value of the annuity. -A sequence of yearly payments of $6000 is invested at the end of each of 11 years at an interest rate of 2.1%, compounded annually. What is the total amount of the annuity after 11 years?

Find the future value of the annuity. -To save for retirement, you decide to deposit $2250 into an IRA at the end of each year for the next 40 years. If the interest rate is 2% per year compounded annually, find the value of the IRA, Rounded to the nearest dollar, after 40 years.

Provide an appropriate response. -The sum of an infinite geometric series exists if:

Find the future value of the annuity. -There are payments of $100 at the end of each year for 11 years at 3% interest compounded annually

Provide an appropriate response. -Consider the sequence defined by a an=−80n−58a _ { n } = - 80 n - 58an​=−80n−58 . Is this sequence arithmetic, geometric, or neither?

Provide an appropriate response. -Convert the decimal .868686….868686 \ldots.868686… to a fraction by writing it as the infinite series 86100+861002+861003+…\frac { 86 } { 100 } + \frac { 86 } { 100 ^ { 2 } } + \frac { 86 } { 100 ^ { 3 } } + \ldots10086​+100286​+100386​+…

Provide an appropriate response. -What is the common ratio in the geometric sequence an=log⁡[6(5n−1) ]\left. a _ { n } = \log \left[ 6 ^ { \left( 5 ^ { n - 1 } \right.} \right) \right]an​=log[6(5n−1) ] ?

Provide an appropriate response. -For 1413,x,811\frac { 14 } { 13 } , x , \frac { 8 } { 11 }1314​,x,118​ to be a geometric sequence, xxx must be:

Evaluate the expression. - 9C2{ } _ { 9 } \mathrm { C } _ { 2 }9​C2​

Provide an appropriate response. -Let f(x) =8(6x) f ( x ) = 8 \left( 6 ^ { x } \right) f(x) =8(6x) , and consider the sequence f(1) ,f(2) ,f(3) ,…f ( 1 ) , f ( 2 ) , f ( 3 ) , \ldotsf(1) ,f(2) ,f(3) ,… Is this a geometric sequence, and if so, what is the common ratio?

Evaluate the expression. - (1293) \left( \begin{array} { c } 129 \\3\end{array} \right) (1293​)

Evaluate the expression. - (103) \left( \begin{array} { c } 10 \\3\end{array} \right) (103​)

Evaluate the expression. - 7!5!2!\frac { 7 ! } { 5 ! 2 ! }5!2!7!​

Provide an appropriate response. -For 7,x,57 , x , 57,x,5 to be an arithmetic sequence, xxx must be:

Evaluate the expression. - 10!5!5!\frac { 10 ! } { 5 ! 5 ! }5!5!10!​