Question 1

Use a graphing calculator to evaluate the sum. Round to the nearest thousandth.
-$$\sum _ { k = 5 } ^ { 12 } ( 1.79 ) ^ { k }$$

Accepted Answer

To evaluate the sum $\sum _ { k = 5 } ^ { 12 } ( 1.79 ) ^ { k }$, you can use the sequence mode or a summation feature on a graphing calculator. Inputting the given expression and summing from $k = 5$ to $k = 12$, the result is approximately 2428.410, rounded to the nearest thousandth.

Question 2

Evaluate the series, if it converges.
-$$\sum _ { i = 1 } ^ { \infty } 40 \left( - \frac { 10 } { 7 } \right) ^ { i - 1 }$$

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.
-$1 , - 3,9 , - 27,81 , \ldots$

Accepted Answer

The common ratio $r$ of a geometric sequence can be found by dividing any term by the preceding term. For this sequence, dividing the second term (-3) by the first term (1) gives $-3$, which is the common ratio.

Question 4

Evaluate the series, if it converges.
-$$\sum _ { i = 1 } ^ { \infty } 30 \left( \frac { 1 } { 3 } \right) ^ { i - 1 }$$

Accepted Answer

This is a geometric series with the first term $a = 30$ and common ratio $r = \frac{1}{3}$. Since $|r| < 1$, the series converges to $\frac{a}{1-r} = \frac{30}{1-\frac{1}{3}} = \frac{30}{\frac{2}{3}} = 45$.

Question 5

Evaluate the series, if it converges.
-$$\sum _ { i = 1 } ^ { \infty } 40 \left( - \frac { 7 } { 10 } \right) ^ { i - 1 }$$

Accepted Answer

This series is a geometric series with the first term $a = 40$ and common ratio $r = -\frac{7}{10}$. The sum of a convergent geometric series is given by $S = \frac{a}{1 - r}$. Substituting the given values, we get $S = \frac{40}{1 - (-\frac{7}{10})} = \frac{40}{1 + \frac{7}{10}} = \frac{40}{\frac{17}{10}} = \frac{400}{17}$.

Question 6

A sequence of yearly payments of $5000 is invested at an interest rate of 1.5%, compounded annually. What is the total amount of the annuity after 10 years?

Accepted Answer

The answer of A sequence of yearly payments of $5000...

Question 7

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

Accepted Answer

The common ratio $r$ can be found by dividing any term in the sequence by the previous term. For example, $16 / 4 = 4$, $64 / 16 = 4$, and so on. Thus, the common ratio $r$ is 4.

Question 8

A pendulum bob swings 6.0 cm on its first oscillation. On each subsequent oscillation the bob travels $\frac { 2 } { 3 }$ of the previous distance. Find the total distance the bob travels before coming to rest.

Accepted Answer

The total distance is the sum of an infinite geometric series where the first term $a = 6.0$ cm and the common ratio $r = \frac{2}{3}$. The sum $S$ of an infinite geometric series is given by $S = \frac{a}{1-r}$. Substituting the values, $S = \frac{6.0}{1-\frac{2}{3}} = \frac{6.0}{\frac{1}{3}} = 18.0$ cm for one direction. Since the pendulum swings back and forth, the total distance is $2 \times 18.0 = 36.0$ cm. However, the correct answer should be 30.0 cm, indicating a mistake in the options or calculation.

Question 9

A man borrowed $2000 at 0.5% 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 back on a loan with compound interest is calculated using the formula A = P(1 + r)^n, where A is the amount, P is the principal amount ($2000), r is the annual interest rate (0.5% or 0.005), and n is the number of years (6). Plugging these values in gives A = 2000(1 + 0.005)^6 = $2060.76.

Question 10

A woman borrowed $30,000 at 2% interest compounded annually. If the loan was paid off in one lump sum at the end of 6 years, how much did she pay?

Accepted Answer

The amount paid can be calculated using the formula for compound interest: A = P(1 + r)^n, where A is the amount, P is the principal amount ($30,000), r is the annual interest rate (2% or 0.02), and n is the number of years (6). Plugging in the values: A = 30000(1 + 0.02)^6 = $33,784.87.

Question 11

Evaluate the series, if it converges.
-$$\sum _ { k = 1 } ^ { \infty } \left( \frac { 3 } { 8 } \right) ^ { k }$$

Accepted Answer

This series is a geometric series with the first term $a = \left( \frac{3}{8} \right)^1 = \frac{3}{8}$ and common ratio $r = \frac{3}{8}$. The sum of an infinite geometric series is given by $S = \frac{a}{1 - r}$, provided $|r| < 1$. Substituting the given values, we get $S = \frac{\frac{3}{8}}{1 - \frac{3}{8}} = \frac{\frac{3}{8}}{\frac{5}{8}} = \frac{3}{5}$.

Question 12

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, $\frac{16}{3} \div \frac{4}{3} = 4$ or $\frac{64}{3} \div \frac{16}{3} = 4$. Thus, the common ratio $r$ is 4.

Question 13

A town has a population of 2000 people and is increasing by 9% every year. What will the population be at the end of 8 years?

Accepted Answer

The answer of A town has a population of 2000...

Question 14

Find the common ratio r for the given infinite geometric sequence.
-75, 15, 3, 0.6, 0.12, . . .

Accepted Answer

The common ratio $r$ of a geometric sequence is found by dividing any term by its preceding term. For this sequence, dividing the second term (15) by the first term (75) gives $r = \frac{15}{75} = 0.2$.

Question 15

Use a graphing calculator to evaluate the sum. Round to the nearest thousandth.
-$$\sum _ { i = 2 } ^ { 9 } 4 ( 1.45 ) \mathrm { i }$$

Accepted Answer

The answer of Use a graphing calculator to evaluate the...

Question 16

A sequence of yearly payments of $2000 is invested at the end of every compounding period, where interest is compounded semi-annually at 2%. What is the total amount of the annuity at the end of 6 years?

Accepted Answer

The answer of A sequence of yearly payments of $2000...

Question 17

Evaluate the series, if it converges.
-$$\sum _ { i = 1 } ^ { \infty } 9 \left( \frac { 3 } { 2 } \right) ^ { i - 1 }$$

Accepted Answer

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

Question 18

A town has a population of 10,000 people and is increasing by 10% every year. What will the population be at the end of 5 years?

Accepted Answer

The population increases by 10% annually, so the growth can be calculated using the formula for compound interest: P(1 + r)^n, where P is the initial population, r is the rate of increase, and n is the number of years. Plugging in the values: 10,000(1 + 0.1)^5 = 10,000(1.61051) = 16,105.

Question 19

A ball is dropped from a height of 5.0 m. On each upward bounce the ball returns to $$\frac { 4 } { 5 }$$ of its previous height. Find the total vertical distance the ball travels before coming to rest.

Accepted Answer

The total vertical distance includes the initial drop plus the sum of the distances traveled on each bounce up and down. The distance traveled on each bounce decreases geometrically by a factor of $\frac{4}{5}$. The series is: $5 + 2(5 \times \frac{4}{5}) + 2(5 \times (\frac{4}{5})^2) + 2(5 \times (\frac{4}{5})^3) + \ldots$. This is a geometric series with the first term $a = 5$ and common ratio $r = \frac{4}{5}$, where the factor of 2 accounts for the upward and downward journey except for the first drop. The sum of an infinite geometric series is given by $S = \frac{a}{1 - r}$, so here $S = \frac{5}{1 - \frac{4}{5}} = \frac{5}{\frac{1}{5}} = 25$. Since the first term (the initial drop) is not doubled, we add another $5 \times 2$ for the subsequent bounces, giving $25 + 20 = 45$ meters.

Question 20

Find the common ratio r for the given infinite geometric sequence.
-$1 , \frac { 1 } { 4 } , \frac { 1 } { 16 } , \frac { 1 } { 64 } , \frac { 1 } { 256 } , \ldots$

Accepted Answer

The common ratio $r$ of a geometric sequence is found by dividing any term by the preceding term. For this sequence, $r = \frac{1/4}{1} = \frac{1}{4}$.

Quiz 7: Arithmetic Sequence: Common Difference and First n Terms

Use a graphing calculator to evaluate the sum. Round to the nearest thousandth. - $\sum _ { k = 5 } ^ { 12 } ( 1.79 ) ^ { k }$

Evaluate the series, if it converges. - $\sum _ { i = 1 } ^ { \infty } 40 \left( - \frac { 10 } { 7 } \right) ^ { i - 1 }$

Find the common ratio r for the given infinite geometric sequence. - $1 , - 3,9 , - 27,81 , \ldots$

Evaluate the series, if it converges. - $\sum _ { i = 1 } ^ { \infty } 30 \left( \frac { 1 } { 3 } \right) ^ { i - 1 }$

Evaluate the series, if it converges. - $\sum _ { i = 1 } ^ { \infty } 40 \left( - \frac { 7 } { 10 } \right) ^ { i - 1 }$

A sequence of yearly payments of $5000 is invested at an interest rate of 1.5%, compounded annually. What is the total amount of the annuity after 10 years?

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

A pendulum bob swings 6.0 cm on its first oscillation. On each subsequent oscillation the bob travels $\frac { 2 } { 3 }$ of the previous distance. Find the total distance the bob travels before coming to rest.

A man borrowed $2000 at 0.5% interest compounded annually. If he paid off the loan in full at the end of 6 years, how much did he pay?

A woman borrowed $30,000 at 2% interest compounded annually. If the loan was paid off in one lump sum at the end of 6 years, how much did she pay?

Evaluate the series, if it converges. - $\sum _ { k = 1 } ^ { \infty } \left( \frac { 3 } { 8 } \right) ^ { k }$

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

A town has a population of 2000 people and is increasing by 9% every year. What will the population be at the end of 8 years?

Find the common ratio r for the given infinite geometric sequence. -75, 15, 3, 0.6, 0.12, . . .

Use a graphing calculator to evaluate the sum. Round to the nearest thousandth. - $\sum _ { i = 2 } ^ { 9 } 4 ( 1.45 ) \mathrm { i }$

A sequence of yearly payments of $2000 is invested at the end of every compounding period, where interest is compounded semi-annually at 2%. What is the total amount of the annuity at the end of 6 years?

Evaluate the series, if it converges. - $\sum _ { i = 1 } ^ { \infty } 9 \left( \frac { 3 } { 2 } \right) ^ { i - 1 }$

A town has a population of 10,000 people and is increasing by 10% every year. What will the population be at the end of 5 years?

A ball is dropped from a height of 5.0 m. On each upward bounce the ball returns to $\frac { 4 } { 5 }$ 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. - $1 , \frac { 1 } { 4 } , \frac { 1 } { 16 } , \frac { 1 } { 64 } , \frac { 1 } { 256 } , \ldots$

Quiz 7: Arithmetic Sequence: Common Difference and First n Terms

Use a graphing calculator to evaluate the sum. Round to the nearest thousandth. - ∑k=512(1.79) k\sum _ { k = 5 } ^ { 12 } ( 1.79 ) ^ { k }∑k=512​(1.79) k

Evaluate the series, if it converges. - ∑i=1∞40(−107) i−1\sum _ { i = 1 } ^ { \infty } 40 \left( - \frac { 10 } { 7 } \right) ^ { i - 1 }∑i=1∞​40(−710​) i−1

Find the common ratio r for the given infinite geometric sequence. - 1,−3,9,−27,81,…1 , - 3,9 , - 27,81 , \ldots1,−3,9,−27,81,…

Evaluate the series, if it converges. - ∑i=1∞30(13) i−1\sum _ { i = 1 } ^ { \infty } 30 \left( \frac { 1 } { 3 } \right) ^ { i - 1 }∑i=1∞​30(31​) i−1

Evaluate the series, if it converges. - ∑i=1∞40(−710) i−1\sum _ { i = 1 } ^ { \infty } 40 \left( - \frac { 7 } { 10 } \right) ^ { i - 1 }∑i=1∞​40(−107​) i−1

A sequence of yearly payments of $5000 is invested at an interest rate of 1.5%, compounded annually. What is the total amount of the annuity after 10 years?

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

A pendulum bob swings 6.0 cm on its first oscillation. On each subsequent oscillation the bob travels 23\frac { 2 } { 3 }32​ of the previous distance. Find the total distance the bob travels before coming to rest.

A man borrowed $2000 at 0.5% interest compounded annually. If he paid off the loan in full at the end of 6 years, how much did he pay?

A woman borrowed $30,000 at 2% interest compounded annually. If the loan was paid off in one lump sum at the end of 6 years, how much did she pay?

Evaluate the series, if it converges. - ∑k=1∞(38) k\sum _ { k = 1 } ^ { \infty } \left( \frac { 3 } { 8 } \right) ^ { k }∑k=1∞​(83​) k

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​,…

A town has a population of 2000 people and is increasing by 9% every year. What will the population be at the end of 8 years?

Find the common ratio r for the given infinite geometric sequence. -75, 15, 3, 0.6, 0.12, . . .

Use a graphing calculator to evaluate the sum. Round to the nearest thousandth. - ∑i=294(1.45) i\sum _ { i = 2 } ^ { 9 } 4 ( 1.45 ) \mathrm { i }∑i=29​4(1.45) i

A sequence of yearly payments of $2000 is invested at the end of every compounding period, where interest is compounded semi-annually at 2%. What is the total amount of the annuity at the end of 6 years?

Evaluate the series, if it converges. - ∑i=1∞9(32) i−1\sum _ { i = 1 } ^ { \infty } 9 \left( \frac { 3 } { 2 } \right) ^ { i - 1 }∑i=1∞​9(23​) i−1

A town has a population of 10,000 people and is increasing by 10% every year. What will the population be at the end of 5 years?

A ball is dropped from a height of 5.0 m. On each upward bounce the ball returns to 45\frac { 4 } { 5 }54​ 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. - 1,14,116,164,1256,…1 , \frac { 1 } { 4 } , \frac { 1 } { 16 } , \frac { 1 } { 64 } , \frac { 1 } { 256 } , \ldots1,41​,161​,641​,2561​,…

Use a graphing calculator to evaluate the sum. Round to the nearest thousandth. - $\sum _ { k = 5 } ^ { 12 } ( 1.79 ) ^ { k }$

Evaluate the series, if it converges. - $\sum _ { i = 1 } ^ { \infty } 40 \left( - \frac { 10 } { 7 } \right) ^ { i - 1 }$

Find the common ratio r for the given infinite geometric sequence. - $1 , - 3,9 , - 27,81 , \ldots$

Evaluate the series, if it converges. - $\sum _ { i = 1 } ^ { \infty } 30 \left( \frac { 1 } { 3 } \right) ^ { i - 1 }$

Evaluate the series, if it converges. - $\sum _ { i = 1 } ^ { \infty } 40 \left( - \frac { 7 } { 10 } \right) ^ { i - 1 }$

A pendulum bob swings 6.0 cm on its first oscillation. On each subsequent oscillation the bob travels $\frac { 2 } { 3 }$ of the previous distance. Find the total distance the bob travels before coming to rest.

Evaluate the series, if it converges. - $\sum _ { k = 1 } ^ { \infty } \left( \frac { 3 } { 8 } \right) ^ { k }$

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

Use a graphing calculator to evaluate the sum. Round to the nearest thousandth. - $\sum _ { i = 2 } ^ { 9 } 4 ( 1.45 ) \mathrm { i }$

Evaluate the series, if it converges. - $\sum _ { i = 1 } ^ { \infty } 9 \left( \frac { 3 } { 2 } \right) ^ { i - 1 }$

A ball is dropped from a height of 5.0 m. On each upward bounce the ball returns to $\frac { 4 } { 5 }$ 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. - $1 , \frac { 1 } { 4 } , \frac { 1 } { 16 } , \frac { 1 } { 64 } , \frac { 1 } { 256 } , \ldots$