Question 1

Use Pascal's triangle to expand the binomial.
-$$( x - 4 ) ^ { 4 }$$

Accepted Answer

The correct expansion using Pascal's triangle for the coefficients (1, 4, 6, 4, 1) is $x^4 - 16x^3 + 96x^2 - 256x + 256$.

Question 2

Solve the problem.
-A job pays a salary of 27,000 the first year. During the next 9 years, the salary increases by 5% each year. What is the salary for the 10th year? What is the total salary over the 10-year period? (Round to the nearest cent.)

Accepted Answer

The salary for the 10th year is calculated using the formula for compound interest: $P(1 + r)^n$, where $P = 27,000$, $r = 0.05$, and $n = 9$. This gives $27,000(1 + 0.05)^9 = 41,885.86$. The total salary over the 10-year period is the sum of a geometric series, which can be calculated using the formula for the sum of a geometric series: $S_n = P \frac{(1 - r^n)}{1 - r}$, where $P = 27,000$, $r = 1.05$, and $n = 10$. This gives a total of $339,603.10$.

Question 3

Solve the problem.
-Lonnie deposits $100 each month into an account paying annual interest of 7% compounded monthly. How much will his account have in it at the end of 10 years? Round to the nearest dollar.

Accepted Answer

The answer of Solve the problem.
-Lonnie deposits $100 each month...

Question 4

Solve the problem.
-Rewrite the number $0 . \overline { 35 }$ as the quotient of two integers.

Accepted Answer

To convert the repeating decimal $0.\overline{35}$ to a fraction, let $x = 0.\overline{35}$. Multiplying both sides by 100 gives $100x = 35.\overline{35}$. Subtracting the original equation from this, $99x = 35$, so $x = \frac{35}{99}$.

Question 5

Determine if the infinite geometric series converges or diverges. If the series converges, find its sum.
-$$3 - \frac { 3 } { 4 } + \frac { 3 } { 16 } - \cdots$$

Accepted Answer

The series is a geometric series with the first term $a = 3$ and common ratio $r = -\frac{1}{4}$. The sum of an infinite geometric series is given by $S = \frac{a}{1 - r}$ if $|r| < 1$. Here, $S = \frac{3}{1 - (-\frac{1}{4})} = \frac{3}{\frac{5}{4}} = \frac{3 \times 4}{5} = \frac{12}{5}$, so the series converges to $\frac{12}{5}$.

Question 6

Determine if the infinite geometric series converges or diverges. If the series converges, find its sum.
-$$45 + 15 + 5 + \frac { 5 } { 3 } + \ldots$$

Accepted Answer

The common ratio is $\frac { 1 } { 3 }$, which is between -1 and 1, so the series converges. The sum is $\frac { 45 } { 1 - \frac { 1 } { 3 } } = \frac { 135 } { 2 }$.

Question 7

Determine if the infinite geometric series converges or diverges. If the series converges, find its sum.
-$$- 6 - 2 - \frac { 2 } { 3 } - \ldots$$

Accepted Answer

The series is geometric with the first term $a = -6$ and common ratio $r = \frac{1}{3}$. Since $|r| < 1$, the series converges. The sum is $S = \frac{a}{1-r} = \frac{-6}{1-\frac{1}{3}} = -9$.

Question 8

Solve the problem.
-On a gambling boat, Gertrude tripled her bet each time she won. If her first winning bet was $4 and she won six consecutive bets, find how much she won on the sixth bet. Find the total amount she won on these six bets.

Accepted Answer

The answer of Solve the problem.
-On a gambling boat, Gertrude...

Question 9

Solve the problem.
-A pendulum bob swings through an arc 50 inches long on its first swing. Each swing thereafter, it swings only 70% as far as on the previous swing. How far will it swing altogether before coming to a complete stop?

Accepted Answer

This problem describes a geometric series where the first term $a = 50$ inches and the common ratio $r = 0.7$. The sum $S$ of an infinite geometric series is given by $S = \frac{a}{1 - r}$, where $a$ is the first term and $r$ is the common ratio. Substituting the given values, $S = \frac{50}{1 - 0.7} = \frac{50}{0.3} = 166.67$ inches, which rounds to 167 inches, making option C correct.

Question 10

Find the sum of the geometric series.
-$\sum _ { i = 1 } ^ { 10 } 400 ( 1.03 ) ^ { i } - 1 \quad$ Round the answer to four decimal places when necessary.

Accepted Answer

The answer of Find the sum of the geometric series.
-\(\sum...

Question 11

Determine if the infinite geometric series converges or diverges. If the series converges, find its sum.
-$$5 - \frac { 5 } { 3 } + \frac { 5 } { 9 } - \frac { 5 } { 27 } + \ldots$$

Accepted Answer

The answer of Determine if the infinite geometric series converges...

Question 12

Solve the problem.
-Looking ahead to retirement, you sign up for automatic savings in a fixed-income 401K plan that pays 6.5% per year compounded annually. You plan to invest $2000 at the end of each year for the next 20 years. How much will your account have in it at the end of 20 years? Round to the nearest dollar.

Accepted Answer

The answer of Solve the problem.
-Looking ahead to retirement, you...

Question 13

Solve the problem.
-When students at the local university held a food drive, 1458 cans of food were collected on the first day of the drive, 486 the second day, 162 the third day, and so on. Find the total number of cans collected the first five days.

Accepted Answer

The answer of Solve the problem.
-When students at the local...

Question 14

Determine if the infinite geometric series converges or diverges. If the series converges, find its sum.
-$$\sum _ { i = 1 } ^ { \infty } 4 \left( \frac { 2 } { 3 } \right) ^ { \mathrm { i } - 1 }$$

Accepted Answer

The answer of Determine if the infinite geometric series converges...

Question 15

Use Pascal's triangle to expand the binomial.
-$( x - 9 ) ^ { 5 }$

Accepted Answer

Using Pascal's triangle, the coefficients for the expansion of $(x - 9)^5$ are 1, 5, 10, 10, 5, 1. The expansion is $x^5 - 45x^4 + 810x^3 - 7290x^2 + 32805x - 59049$.

Question 16

Solve the problem.
-Rewrite the number $0 . \overline { 285 }$ as the quotient of two integers.

Accepted Answer

To convert a repeating decimal to a fraction, you can use the formula $x = 0.\overline{abc}$ to get $1000x = abc.\overline{abc}$ and then subtract $x$ from $1000x$ to isolate the repeating part. For $0.\overline{285}$, let $x = 0.\overline{285}$, then $1000x = 285.\overline{285}$. Subtracting $x$ from $1000x$ gives $999x = 285$, so $x = \frac{285}{999}$. Simplifying this fraction gives $x = \frac{95}{333}$, which matches option B.

Question 17

Solve the problem.
-Ms. Patterson proposes to give her daughter Claire an allowance of $0.10 on the first day of her 12-day vacation, $0.20 on the second day, $0.40 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 $0.10. On the 12th day, the allowance would be $0.10 * 2^(12-1) = $0.10 * 2^11 = $0.10 * 2048 = $204.80.

Question 18

Determine if the infinite geometric series converges or diverges. If the series converges, find its sum.
-$$\sum _ { i = 1 } ^ { \infty } \frac { 1 } { 5 } ( 4 ) ^ { \mathrm { i } } - 1$$

Accepted Answer

The given series is divergent because the common ratio $r = 4 > 1$, and for a geometric series to converge, the absolute value of the common ratio must be less than 1.

Question 19

Solve the problem.
-After being struck with a hammer, a gong vibrates 21 vibrations in the first second and in each second thereafter makes $$\frac { 3 } { 4 }$$ as many vibrations as in the previous second. Find how many vibrations the gong makes before it stops vibrating.

Accepted Answer

The gong's vibrations form a geometric series with the first term $a = 21$ and common ratio $r = \frac{3}{4}$. The series continues until the vibrations stop, effectively an infinite series since the gong never truly stops vibrating but approaches zero. The sum $S$ of an infinite geometric series is given by $S = \frac{a}{1 - r}$, where $a$ is the first term and $r$ is the common ratio. Substituting $a = 21$ and $r = \frac{3}{4}$, we get $S = \frac{21}{1 - \frac{3}{4}} = \frac{21}{\frac{1}{4}} = 21 \times 4 = 84$.

Question 20

Determine if the infinite geometric series converges or diverges. If the series converges, find its sum.
-$$4 + \frac { 4 } { 5 } + \frac { 4 } { 25 } + \frac { 4 } { 125 } + \ldots$$

Accepted Answer

The answer of Determine if the infinite geometric series converges...

Quiz 9: Sequences and Series; Counting and Probability

Use Pascal's triangle to expand the binomial. - $( x - 4 ) ^ { 4 }$

Solve the problem. -A job pays a salary of 27,000 the first year. During the next 9 years, the salary increases by 5% each year. What is the salary for the 10th year? What is the total salary over the 10-year period? (Round to the nearest cent.)

Solve the problem. -Lonnie deposits $100 each month into an account paying annual interest of 7% compounded monthly. How much will his account have in it at the end of 10 years? Round to the nearest dollar.

Solve the problem. -Rewrite the number $0 . \overline { 35 }$ as the quotient of two integers.

Determine if the infinite geometric series converges or diverges. If the series converges, find its sum. - $3 - \frac { 3 } { 4 } + \frac { 3 } { 16 } - \cdots$

Determine if the infinite geometric series converges or diverges. If the series converges, find its sum. - $45 + 15 + 5 + \frac { 5 } { 3 } + \ldots$

Determine if the infinite geometric series converges or diverges. If the series converges, find its sum. - $- 6 - 2 - \frac { 2 } { 3 } - \ldots$

Solve the problem. -On a gambling boat, Gertrude tripled her bet each time she won. If her first winning bet was $4 and she won six consecutive bets, find how much she won on the sixth bet. Find the total amount she won on these six bets.

Solve the problem. -A pendulum bob swings through an arc 50 inches long on its first swing. Each swing thereafter, it swings only 70% as far as on the previous swing. How far will it swing altogether before coming to a complete stop?

Find the sum of the geometric series. - $\sum _ { i = 1 } ^ { 10 } 400 ( 1.03 ) ^ { i } - 1 \quad$ Round the answer to four decimal places when necessary.

Determine if the infinite geometric series converges or diverges. If the series converges, find its sum. - $5 - \frac { 5 } { 3 } + \frac { 5 } { 9 } - \frac { 5 } { 27 } + \ldots$

Solve the problem. -When students at the local university held a food drive, 1458 cans of food were collected on the first day of the drive, 486 the second day, 162 the third day, and so on. Find the total number of cans collected the first five days.

Determine if the infinite geometric series converges or diverges. If the series converges, find its sum. - $\sum _ { i = 1 } ^ { \infty } 4 \left( \frac { 2 } { 3 } \right) ^ { \mathrm { i } - 1 }$

Use Pascal's triangle to expand the binomial. - $( x - 9 ) ^ { 5 }$

Solve the problem. -Rewrite the number $0 . \overline { 285 }$ as the quotient of two integers.

Solve the problem. -Ms. Patterson proposes to give her daughter Claire an allowance of $0.10 on the first day of her 12-day vacation, $0.20 on the second day, $0.40 on the third day, and so on. Find the allowance Claire would receive on the last day of her vacation.

Determine if the infinite geometric series converges or diverges. If the series converges, find its sum. - $\sum _ { i = 1 } ^ { \infty } \frac { 1 } { 5 } ( 4 ) ^ { \mathrm { i } } - 1$

Solve the problem. -After being struck with a hammer, a gong vibrates 21 vibrations in the first second and in each second thereafter makes $\frac { 3 } { 4 }$ as many vibrations as in the previous second. Find how many vibrations the gong makes before it stops vibrating.

Determine if the infinite geometric series converges or diverges. If the series converges, find its sum. - $4 + \frac { 4 } { 5 } + \frac { 4 } { 25 } + \frac { 4 } { 125 } + \ldots$

Quiz 9: Sequences and Series; Counting and Probability

Use Pascal's triangle to expand the binomial. - (x−4) 4( x - 4 ) ^ { 4 }(x−4) 4

Solve the problem. -A job pays a salary of 27,000 the first year. During the next 9 years, the salary increases by 5% each year. What is the salary for the 10th year? What is the total salary over the 10-year period? (Round to the nearest cent.)

Solve the problem. -Lonnie deposits $100 each month into an account paying annual interest of 7% compounded monthly. How much will his account have in it at the end of 10 years? Round to the nearest dollar.

Solve the problem. -Rewrite the number 0.35‾0 . \overline { 35 }0.35 as the quotient of two integers.

Determine if the infinite geometric series converges or diverges. If the series converges, find its sum. - 3−34+316−⋯3 - \frac { 3 } { 4 } + \frac { 3 } { 16 } - \cdots3−43​+163​−⋯

Determine if the infinite geometric series converges or diverges. If the series converges, find its sum. - 45+15+5+53+…45 + 15 + 5 + \frac { 5 } { 3 } + \ldots45+15+5+35​+…

Determine if the infinite geometric series converges or diverges. If the series converges, find its sum. - −6−2−23−…- 6 - 2 - \frac { 2 } { 3 } - \ldots−6−2−32​−…

Solve the problem. -On a gambling boat, Gertrude tripled her bet each time she won. If her first winning bet was $4 and she won six consecutive bets, find how much she won on the sixth bet. Find the total amount she won on these six bets.

Solve the problem. -A pendulum bob swings through an arc 50 inches long on its first swing. Each swing thereafter, it swings only 70% as far as on the previous swing. How far will it swing altogether before coming to a complete stop?

Find the sum of the geometric series. - ∑i=110400(1.03) i−1\sum _ { i = 1 } ^ { 10 } 400 ( 1.03 ) ^ { i } - 1 \quad∑i=110​400(1.03) i−1 Round the answer to four decimal places when necessary.

Determine if the infinite geometric series converges or diverges. If the series converges, find its sum. - 5−53+59−527+…5 - \frac { 5 } { 3 } + \frac { 5 } { 9 } - \frac { 5 } { 27 } + \ldots5−35​+95​−275​+…

Solve the problem. -When students at the local university held a food drive, 1458 cans of food were collected on the first day of the drive, 486 the second day, 162 the third day, and so on. Find the total number of cans collected the first five days.

Determine if the infinite geometric series converges or diverges. If the series converges, find its sum. - ∑i=1∞4(23) i−1\sum _ { i = 1 } ^ { \infty } 4 \left( \frac { 2 } { 3 } \right) ^ { \mathrm { i } - 1 }∑i=1∞​4(32​) i−1

Use Pascal's triangle to expand the binomial. - (x−9) 5( x - 9 ) ^ { 5 }(x−9) 5

Solve the problem. -Rewrite the number 0.285‾0 . \overline { 285 }0.285 as the quotient of two integers.

Solve the problem. -Ms. Patterson proposes to give her daughter Claire an allowance of $0.10 on the first day of her 12-day vacation, $0.20 on the second day, $0.40 on the third day, and so on. Find the allowance Claire would receive on the last day of her vacation.

Determine if the infinite geometric series converges or diverges. If the series converges, find its sum. - ∑i=1∞15(4) i−1\sum _ { i = 1 } ^ { \infty } \frac { 1 } { 5 } ( 4 ) ^ { \mathrm { i } } - 1∑i=1∞​51​(4) i−1

Solve the problem. -After being struck with a hammer, a gong vibrates 21 vibrations in the first second and in each second thereafter makes 34\frac { 3 } { 4 }43​ as many vibrations as in the previous second. Find how many vibrations the gong makes before it stops vibrating.

Determine if the infinite geometric series converges or diverges. If the series converges, find its sum. - 4+45+425+4125+…4 + \frac { 4 } { 5 } + \frac { 4 } { 25 } + \frac { 4 } { 125 } + \ldots4+54​+254​+1254​+…

Use Pascal's triangle to expand the binomial. - $( x - 4 ) ^ { 4 }$

Solve the problem. -Rewrite the number $0 . \overline { 35 }$ as the quotient of two integers.

Determine if the infinite geometric series converges or diverges. If the series converges, find its sum. - $3 - \frac { 3 } { 4 } + \frac { 3 } { 16 } - \cdots$

Determine if the infinite geometric series converges or diverges. If the series converges, find its sum. - $45 + 15 + 5 + \frac { 5 } { 3 } + \ldots$

Determine if the infinite geometric series converges or diverges. If the series converges, find its sum. - $- 6 - 2 - \frac { 2 } { 3 } - \ldots$

Find the sum of the geometric series. - $\sum _ { i = 1 } ^ { 10 } 400 ( 1.03 ) ^ { i } - 1 \quad$ Round the answer to four decimal places when necessary.

Determine if the infinite geometric series converges or diverges. If the series converges, find its sum. - $5 - \frac { 5 } { 3 } + \frac { 5 } { 9 } - \frac { 5 } { 27 } + \ldots$

Determine if the infinite geometric series converges or diverges. If the series converges, find its sum. - $\sum _ { i = 1 } ^ { \infty } 4 \left( \frac { 2 } { 3 } \right) ^ { \mathrm { i } - 1 }$

Use Pascal's triangle to expand the binomial. - $( x - 9 ) ^ { 5 }$

Solve the problem. -Rewrite the number $0 . \overline { 285 }$ as the quotient of two integers.

Determine if the infinite geometric series converges or diverges. If the series converges, find its sum. - $\sum _ { i = 1 } ^ { \infty } \frac { 1 } { 5 } ( 4 ) ^ { \mathrm { i } } - 1$

Solve the problem. -After being struck with a hammer, a gong vibrates 21 vibrations in the first second and in each second thereafter makes $\frac { 3 } { 4 }$ as many vibrations as in the previous second. Find how many vibrations the gong makes before it stops vibrating.

Determine if the infinite geometric series converges or diverges. If the series converges, find its sum. - $4 + \frac { 4 } { 5 } + \frac { 4 } { 25 } + \frac { 4 } { 125 } + \ldots$