Question 1

Evaluate the series, if it converges.
-$$\sum _ { i = 1 } ^ { \infty } 46 \left( - \frac { 3 } { 10 } \right) ^ { i - 1 }$$

Accepted Answer

This series is a geometric series with the first term $a = 46$ and common ratio $r = -\frac{3}{10}$. The sum of a convergent geometric series is given by $S = \frac{a}{1 - r}$. Substituting $a = 46$ and $r = -\frac{3}{10}$, we get $S = \frac{46}{1 - (-\frac{3}{10})} = \frac{46}{1 + \frac{3}{10}} = \frac{46}{\frac{13}{10}} = \frac{460}{13}$.

Question 2

Evaluate the series, if it converges.
-$$\sum _ { \mathrm { i } = 1 } ^ { \infty } 45 \left( \frac { 9 } { 4 } \right) ^ { \mathrm { i } - 1 }$$

Accepted Answer

The answer of Evaluate the series, if it converges.
-\[\sum _...

Question 3

Find the common ratio r for the given infinite geometric sequence.
-A town has a population of 10,000 people and is increasing by 10% every year. What will the 175) population be at the end of 4 years?

Accepted Answer

The common ratio $r$ for the given geometric sequence is 1.10, as the population increases by 10% each year. To find the population at the end of 4 years, use the formula for the $n$th term of a geometric sequence: $a_n = a_1 \cdot r^{(n-1)}$, where $a_1 = 10,000$, $r = 1.10$, and $n = 4$. Thus, $a_4 = 10,000 \cdot (1.10)^3 = 10,000 \cdot 1.331 = 13,310$. However, the correct calculation should be $a_4 = 10,000 \cdot (1.10)^4 = 10,000 \cdot 1.4641 = 14,641$ people, indicating a mistake in the initial explanation but the correct choice is identified based on the correct calculation.

Question 4

Find the common ratio r for the given infinite geometric sequence.
-4, 16, 64, 256, 1024, . . .

Accepted Answer

The common ratio $r$ of a geometric sequence can be found by dividing any term by its preceding term. For this sequence, dividing the second term (16) by the first term (4) gives $r = 16 / 4 = 4$.

Question 5

Find the common ratio r for the given infinite geometric sequence.
-The beginning population of a small town was 13,000 people. Due to decline in industrial growth the population has since been decreasing at a rate of 5% every year. What was the population of
This town 10 years later?

Accepted Answer

The common ratio $r$ for the given geometric sequence is $1 - 0.05 = 0.95$, since the population decreases by 5% each year. To find the population 10 years later, we use the formula for the $n$th term of a geometric sequence: $a_n = a_1 \cdot r^{(n-1)}$, where $a_1 = 13000$, $r = 0.95$, and $n = 11$ (since we start counting from year 0 for the initial population). Thus, $a_{11} = 13000 \cdot 0.95^{10} \approx 7784$ people.

Question 6

Find the common ratio r for the given infinite geometric sequence.
-A sequence of yearly payments of $1900 is invested at the end of every compounding period, where interest is compounded semi-annually at 2.5%. What is the total amount of the annuity at
The end of 6 years?

Accepted Answer

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

Question 7

Find the common ratio r for the given infinite geometric sequence.
-$$3 , \frac { 3 } { 2 } , \frac { 3 } { 4 } , \frac { 3 } { 8 } , \frac { 3 } { 16 } , \ldots$$

Accepted Answer

The common ratio $r$ can be found by dividing any term in the sequence by the term preceding it. For example, $\frac{3/2}{3} = \frac{1}{2}$, so the common ratio is $\frac{1}{2}$.

Question 8

Evaluate the series, if it converges.
-$$\sum _ { i = 1 } ^ { \infty } 30 \left( - \frac { 7 } { 6 } \right) ^ { i - 1 }$$

Accepted Answer

The answer of Evaluate the series, if it converges.
-\[\sum _...

Question 9

Find the common ratio r for the given infinite geometric sequence.
-A pendulum bob swings 6.0 cm on its first oscillation. On each subsequent oscillation the bob travels $\frac { 1 } { 3 }$ of the previous distance. Find the total distance the bob travels before coming to rest.

Accepted Answer

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

Question 10

Find the common ratio r for the given infinite geometric sequence.
-A man borrowed $3000 at 1.5% interest compounded annually. If he paid off the loan in full at the end of 6 years, how much did he pay?

Accepted Answer

The amount paid off for a loan compounded annually is calculated using the formula $A = P(1 + r)^n$, where $A$ is the amount, $P$ is the principal amount, $r$ is the annual interest rate, and $n$ is the number of years. Here, $P = $3000$, $r = 1.5\% = 0.015$, and $n = 6$. Plugging these values into the formula gives $A = 3000(1 + 0.015)^6$, which calculates to approximately $3280.33.

Question 11

Find the common ratio r for the given infinite geometric sequence.
-3, -9, 27, -81, 243, . . .

Accepted Answer

The common ratio $r$ can be found by dividing any term in the sequence by its preceding term, e.g., $\frac{-9}{3} = -3$ or $\frac{27}{-9} = -3$. This shows that the sequence multiplies by -3 at each step.

Question 12

Use a graphing calculator to evaluate the sum. Round to the nearest thousandth.
-$$\sum _ { \mathrm { k } = 5 } ^ { 10 } ( 0.53 ) \mathrm { k }$$

Accepted Answer

The sum can be calculated by evaluating the expression $0.53k$ for each integer value of $k$ from 5 to 10 and then adding those values together. The correct sum, rounded to the nearest thousandth, is $0.087$.

Question 13

Find the common ratio r for the given infinite geometric sequence.
-A sequence of yearly payments of $3000 is invested at an interest rate of 1.5%, compounded annually. What is the total amount of the annuity after 9 years?

Accepted Answer

The total amount of an annuity can be calculated using the formula for the future value of an annuity: $A = P \frac{(1 + r)^n - 1}{r}$, where $A$ is the amount, $P$ is the payment, $r$ is the interest rate per period, and $n$ is the number of periods. Here, $P = 3000$, $r = 0.015$ (1.5%), and $n = 9$. Plugging these values into the formula gives the total amount as approximately $28,678.00.

Question 14

Find the common ratio r for the given infinite geometric sequence.
-$$\frac { 3 } { 5 }$$ A ball is dropped from a height of 5.0 m. On each upward bounce the ball returns to of its previous height. Find the total vertical distance the ball travels before coming to rest.

Accepted Answer

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

Question 15

Find the common ratio r for the given infinite geometric sequence.
-A town has a population of 1000 people and is increasing by 9% every year. What will the population be at the end of 8 years?

Accepted Answer

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

Question 16

Evaluate the series, if it converges.
-$$\sum _ { \mathrm { k } = 1 } ^ { \infty } \left( \frac { 3 } { 4 } \right) ^ { \mathrm { k } }$$

Accepted Answer

This series is a geometric series with a common ratio $r = \frac{3}{4}$ and the first term $a = \left(\frac{3}{4}\right)^1 = \frac{3}{4}$. A geometric series converges if $|r| < 1$, which is true in this case. The sum of a converging 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, we get $S = \frac{\frac{3}{4}}{1 - \frac{3}{4}} = \frac{\frac{3}{4}}{\frac{1}{4}} = 3$.

Question 17

Find the common ratio r for the given infinite geometric sequence.
-A woman borrowed $10,000 at 6% interest compounded annually. If the loan was paid off in one lump sum at the end of 4 years, how much did she pay?

Accepted Answer

The common ratio r in this context refers to the growth factor of the loan due to interest, which is 1 + 0.06 = 1.06 for a 6% annual interest rate. However, the question actually asks for the final amount paid after 4 years, not the common ratio. The formula for compound interest is A = P(1 + r)^n, where A is the amount, P is the principal amount ($10,000), r is the annual interest rate (0.06), and n is the number of years (4). Plugging these values in gives A = $10,000(1 + 0.06)^4 = $12,624.77.

Question 18

Find the common ratio r for the given infinite geometric sequence.
-$$75,30,12,4.8,1.92 , \ldots$$

Accepted Answer

The common ratio $r$ of a geometric sequence can be found by dividing any term by its preceding term. For this sequence, dividing the second term (30) by the first term (75) gives $30 / 75 = 0.4$, which is the common ratio.

Question 19

Use a graphing calculator to evaluate the sum. Round to the nearest thousandth.
-$$\sum _ { i = 4 } ^ { 10 } 4 ( 2.03 ) \mathrm { i }$$

Accepted Answer

The sum is calculated by evaluating the expression $4(2.03)^i$ for each integer $i$ from 4 to 10 and then summing these values. Using a graphing calculator, this results in approximately 9302.748.

Question 20

Find the common ratio r for the given infinite geometric sequence.
-$$\frac { 4 } { 3 } , \frac { 16 } { 3 } , \frac { 64 } { 3 } , \frac { 256 } { 3 } , \frac { 1024 } { 3 } , \ldots$$

Accepted Answer

The common ratio $r$ can be found by dividing any term in the sequence by the term preceding it. For example, dividing the second term $\frac{16}{3}$ by the first term $\frac{4}{3}$ gives $r = 4$.

Quiz 12: Further Topics in Algebra

Evaluate the series, if it converges. - $\sum _ { i = 1 } ^ { \infty } 46 \left( - \frac { 3 } { 10 } \right) ^ { i - 1 }$

Evaluate the series, if it converges. - $\sum _ { \mathrm { i } = 1 } ^ { \infty } 45 \left( \frac { 9 } { 4 } \right) ^ { \mathrm { i } - 1 }$

Find the common ratio r for the given infinite geometric sequence. -A town has a population of 10,000 people and is increasing by 10% every year. What will the 175) population be at the end of 4 years?

Find the common ratio r for the given infinite geometric sequence. -4, 16, 64, 256, 1024, . . .

Find the common ratio r for the given infinite geometric sequence. -The beginning population of a small town was 13,000 people. Due to decline in industrial growth the population has since been decreasing at a rate of 5% every year. What was the population of This town 10 years later?

Find the common ratio r for the given infinite geometric sequence. -A sequence of yearly payments of $1900 is invested at the end of every compounding period, where interest is compounded semi-annually at 2.5%. What is the total amount of the annuity at The end of 6 years?

Find the common ratio r for the given infinite geometric sequence. - $3 , \frac { 3 } { 2 } , \frac { 3 } { 4 } , \frac { 3 } { 8 } , \frac { 3 } { 16 } , \ldots$

Evaluate the series, if it converges. - $\sum _ { i = 1 } ^ { \infty } 30 \left( - \frac { 7 } { 6 } \right) ^ { i - 1 }$

Find the common ratio r for the given infinite geometric sequence. -A pendulum bob swings 6.0 cm on its first oscillation. On each subsequent oscillation the bob travels $\frac { 1 } { 3 }$ of the previous distance. Find the total distance the bob travels before coming to rest.

Find the common ratio r for the given infinite geometric sequence. -A man borrowed $3000 at 1.5% interest compounded annually. If he paid off the loan in full at the end of 6 years, how much did he pay?

Find the common ratio r for the given infinite geometric sequence. -3, -9, 27, -81, 243, . . .

Use a graphing calculator to evaluate the sum. Round to the nearest thousandth. - $\sum _ { \mathrm { k } = 5 } ^ { 10 } ( 0.53 ) \mathrm { k }$

Find the common ratio r for the given infinite geometric sequence. -A sequence of yearly payments of $3000 is invested at an interest rate of 1.5%, compounded annually. What is the total amount of the annuity after 9 years?

Find the common ratio r for the given infinite geometric sequence. - $\frac { 3 } { 5 }$ A ball is dropped from a height of 5.0 m. On each upward bounce the ball returns to of its previous height. Find the total vertical distance the ball travels before coming to rest.

Find the common ratio r for the given infinite geometric sequence. -A town has a population of 1000 people and is increasing by 9% every year. What will the population be at the end of 8 years?

Evaluate the series, if it converges. - $\sum _ { \mathrm { k } = 1 } ^ { \infty } \left( \frac { 3 } { 4 } \right) ^ { \mathrm { k } }$

Find the common ratio r for the given infinite geometric sequence. -A woman borrowed $10,000 at 6% interest compounded annually. If the loan was paid off in one lump sum at the end of 4 years, how much did she pay?

Find the common ratio r for the given infinite geometric sequence. - $75,30,12,4.8,1.92 , \ldots$

Use a graphing calculator to evaluate the sum. Round to the nearest thousandth. - $\sum _ { i = 4 } ^ { 10 } 4 ( 2.03 ) \mathrm { i }$

Find the common ratio r for the given infinite geometric sequence. - $\frac { 4 } { 3 } , \frac { 16 } { 3 } , \frac { 64 } { 3 } , \frac { 256 } { 3 } , \frac { 1024 } { 3 } , \ldots$

Quiz 12: Further Topics in Algebra

Evaluate the series, if it converges. - ∑i=1∞46(−310) i−1\sum _ { i = 1 } ^ { \infty } 46 \left( - \frac { 3 } { 10 } \right) ^ { i - 1 }∑i=1∞​46(−103​) i−1

Evaluate the series, if it converges. - ∑i=1∞45(94) i−1\sum _ { \mathrm { i } = 1 } ^ { \infty } 45 \left( \frac { 9 } { 4 } \right) ^ { \mathrm { i } - 1 }∑i=1∞​45(49​) i−1

Find the common ratio r for the given infinite geometric sequence. -A town has a population of 10,000 people and is increasing by 10% every year. What will the 175) population be at the end of 4 years?

Find the common ratio r for the given infinite geometric sequence. -4, 16, 64, 256, 1024, . . .

Find the common ratio r for the given infinite geometric sequence. -The beginning population of a small town was 13,000 people. Due to decline in industrial growth the population has since been decreasing at a rate of 5% every year. What was the population of This town 10 years later?

Find the common ratio r for the given infinite geometric sequence. -A sequence of yearly payments of $1900 is invested at the end of every compounding period, where interest is compounded semi-annually at 2.5%. What is the total amount of the annuity at The end of 6 years?

Find the common ratio r for the given infinite geometric sequence. - 3,32,34,38,316,…3 , \frac { 3 } { 2 } , \frac { 3 } { 4 } , \frac { 3 } { 8 } , \frac { 3 } { 16 } , \ldots3,23​,43​,83​,163​,…

Evaluate the series, if it converges. - ∑i=1∞30(−76) i−1\sum _ { i = 1 } ^ { \infty } 30 \left( - \frac { 7 } { 6 } \right) ^ { i - 1 }∑i=1∞​30(−67​) i−1

Find the common ratio r for the given infinite geometric sequence. -A pendulum bob swings 6.0 cm on its first oscillation. On each subsequent oscillation the bob travels 13\frac { 1 } { 3 }31​ of the previous distance. Find the total distance the bob travels before coming to rest.

Find the common ratio r for the given infinite geometric sequence. -A man borrowed $3000 at 1.5% interest compounded annually. If he paid off the loan in full at the end of 6 years, how much did he pay?

Find the common ratio r for the given infinite geometric sequence. -3, -9, 27, -81, 243, . . .

Use a graphing calculator to evaluate the sum. Round to the nearest thousandth. - ∑k=510(0.53) k\sum _ { \mathrm { k } = 5 } ^ { 10 } ( 0.53 ) \mathrm { k }∑k=510​(0.53) k

Find the common ratio r for the given infinite geometric sequence. -A sequence of yearly payments of $3000 is invested at an interest rate of 1.5%, compounded annually. What is the total amount of the annuity after 9 years?

Find the common ratio r for the given infinite geometric sequence. - 35\frac { 3 } { 5 }53​ A ball is dropped from a height of 5.0 m. On each upward bounce the ball returns to of its previous height. Find the total vertical distance the ball travels before coming to rest.

Find the common ratio r for the given infinite geometric sequence. -A town has a population of 1000 people and is increasing by 9% every year. What will the population be at the end of 8 years?

Evaluate the series, if it converges. - ∑k=1∞(34) k\sum _ { \mathrm { k } = 1 } ^ { \infty } \left( \frac { 3 } { 4 } \right) ^ { \mathrm { k } }∑k=1∞​(43​) k

Find the common ratio r for the given infinite geometric sequence. -A woman borrowed $10,000 at 6% interest compounded annually. If the loan was paid off in one lump sum at the end of 4 years, how much did she pay?

Find the common ratio r for the given infinite geometric sequence. - 75,30,12,4.8,1.92,…75,30,12,4.8,1.92 , \ldots75,30,12,4.8,1.92,…

Use a graphing calculator to evaluate the sum. Round to the nearest thousandth. - ∑i=4104(2.03) i\sum _ { i = 4 } ^ { 10 } 4 ( 2.03 ) \mathrm { i }∑i=410​4(2.03) i

Find the common ratio r for the given infinite geometric sequence. - 43,163,643,2563,10243,…\frac { 4 } { 3 } , \frac { 16 } { 3 } , \frac { 64 } { 3 } , \frac { 256 } { 3 } , \frac { 1024 } { 3 } , \ldots34​,316​,364​,3256​,31024​,…

Evaluate the series, if it converges. - $\sum _ { i = 1 } ^ { \infty } 46 \left( - \frac { 3 } { 10 } \right) ^ { i - 1 }$

Evaluate the series, if it converges. - $\sum _ { \mathrm { i } = 1 } ^ { \infty } 45 \left( \frac { 9 } { 4 } \right) ^ { \mathrm { i } - 1 }$

Find the common ratio r for the given infinite geometric sequence. - $3 , \frac { 3 } { 2 } , \frac { 3 } { 4 } , \frac { 3 } { 8 } , \frac { 3 } { 16 } , \ldots$

Evaluate the series, if it converges. - $\sum _ { i = 1 } ^ { \infty } 30 \left( - \frac { 7 } { 6 } \right) ^ { i - 1 }$

Find the common ratio r for the given infinite geometric sequence. -A pendulum bob swings 6.0 cm on its first oscillation. On each subsequent oscillation the bob travels $\frac { 1 } { 3 }$ of the previous distance. Find the total distance the bob travels before coming to rest.

Use a graphing calculator to evaluate the sum. Round to the nearest thousandth. - $\sum _ { \mathrm { k } = 5 } ^ { 10 } ( 0.53 ) \mathrm { k }$

Find the common ratio r for the given infinite geometric sequence. - $\frac { 3 } { 5 }$ A ball is dropped from a height of 5.0 m. On each upward bounce the ball returns to of its previous height. Find the total vertical distance the ball travels before coming to rest.

Evaluate the series, if it converges. - $\sum _ { \mathrm { k } = 1 } ^ { \infty } \left( \frac { 3 } { 4 } \right) ^ { \mathrm { k } }$

Find the common ratio r for the given infinite geometric sequence. - $75,30,12,4.8,1.92 , \ldots$

Use a graphing calculator to evaluate the sum. Round to the nearest thousandth. - $\sum _ { i = 4 } ^ { 10 } 4 ( 2.03 ) \mathrm { i }$

Find the common ratio r for the given infinite geometric sequence. - $\frac { 4 } { 3 } , \frac { 16 } { 3 } , \frac { 64 } { 3 } , \frac { 256 } { 3 } , \frac { 1024 } { 3 } , \ldots$