Question 1

Solve the problem.
-A town has a population of 1000 people and is increasing by 9% every year. What will the population be at the end of 9 years?
A) 1810 people
B) 1993 people
C) 2367 people
D) 2172 people

Accepted Answer

The population after 9 years can be calculated using the formula for compound interest: P = P0(1 + r)^t, where P0 is the initial population, r is the growth rate, and t is the time in years. Here, P0 = 1000, r = 0.09, and t = 9. So, P = 1000(1 + 0.09)^9 ≈ 2172 people.

Question 2

Solve the problem.
-A man borrowed $2000 at 2.5% interest compounded annually. If he paid off the loan in full at the end of 6 years, how much did he pay?
A) $2050.00
B) $2300.00
C) $319.39
D) $2319.39

Accepted Answer

The amount paid after 6 years 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 (2.5% or 0.025), and n is the number of years (6). Plugging in the values, A = 2000(1 + 0.025)^6 = $2319.39.

Question 3

Evaluate the series, if it converges.
-$$\sum _ { \mathrm { k } = 1 } ^ { \infty } \left( \frac { 3 } { 10 } \right) ^ { \mathrm { k } }$$
A) Does not converge.
B) $\frac { 10 } { 7 }$
C) $\frac { 3 } { 7 }$
D) $- \frac { 3 } { 7 }$

Accepted Answer

The series is a geometric series with first term $a = \frac{3}{10}$ and common ratio $r = \frac{3}{10}$. Since $|r| < 1$, the series converges to $\frac{a}{1-r} = \frac{\frac{3}{10}}{1-\frac{3}{10}} = \frac{3}{7}$.

Question 4

Find the common ratio r for the given infinite geometric sequence.
-$$3 , - 9,27 , - 81,243 , \ldots$$
A) 3
B) $- 12$
C) $- \frac { 1 } { 3 }$
D) $- 3$

Accepted Answer

The common ratio $r$ can be found by dividing any term in the sequence by its preceding term. For example, $\frac{-9}{3} = -3$ or $\frac{27}{-9} = -3$. Thus, the common ratio $r$ is $-3$.

Question 5

Use a graphing calculator to evaluate the sum. Round to the nearest thousandth.
-$$\sum _ { i = 2 } ^ { 9 } - 2 ( 0.93 ) i$$
A) $- 10.883$
B) $- 9.154$
C) $5.442$
D) $- 9.843$

Accepted Answer

The sum can be calculated by inputting the expression $-2(0.93)i$ into a graphing calculator and summing from $i = 2$ to $i = 9$. The result is $-10.883$, rounded to the nearest thousandth.

Question 6

Evaluate the series, if it converges.
-$$\sum _ { i = 1 } ^ { \infty } 25 \left( \frac { 7 } { 10 } \right) ^ { i - 1 }$$
A) $\frac { 250 } { 3 }$
B) $\frac { 175 } { 3 }$
C) Does not converge.
D) $\frac { 10 } { 3 }$

Accepted Answer

This series is a geometric series with the first term $a = 25$ and common ratio $r = \frac{7}{10}$. The sum of a convergent geometric series is given by $S = \frac{a}{1 - r}$. Substituting $a = 25$ and $r = \frac{7}{10}$, we get $S = \frac{25}{1 - \frac{7}{10}} = \frac{25}{\frac{3}{10}} = \frac{250}{3}$.

Question 7

Solve the problem.
-A woman borrowed $10,000 at 2% interest compounded annually. If the loan was paid off in one lump sum at the end of 5 years, how much did she pay?
A) $11,000.00
B) $1040.81
C) $11,040.81
D) $10,200.00

Accepted Answer

The amount paid after 5 years with compound interest is calculated using the formula A = P(1 + r)^n, where A is the amount, P is the principal amount ($10,000), r is the annual interest rate (2% or 0.02), and n is the number of years (5). Plugging in the values, A = 10000(1 + 0.02)^5 = $11,040.81.

Question 8

Use a graphing calculator to evaluate the sum. Round to the nearest thousandth.
-$\sum _ { \mathrm { k } = 1 } ^ { 6 } ( 1.41 ) ^ { \mathrm { k } }$
A) $14.317$
B) $15.727$
C) $23.585$
D) $22.175$

Accepted Answer

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

Question 9

Evaluate the series, if it converges.
-$$\sum _ { i = 1 } ^ { \infty } 24 \left( \frac { 7 } { 2 } \right) ^ { i - 1 }$$
A) Does not converge.
B) $- \frac { 48 } { 5 }$
C) $\frac { 168 } { 5 }$
D) $\frac { 48 } { 5 }$

Accepted Answer

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

Question 10

Solve the problem.
-A pendulum bob swings $5.0 \mathrm {~cm}$ on its first oscillation. On each subsequent oscillation the bob travels $\frac { 3 } { 5 }$ of the previous distance. Find the total distance the bob travels before coming to rest.
A) $21.0 \mathrm {~cm}$
B) $12.5 \mathrm {~cm}$
C) $22.2 \mathrm {~cm}$
D) $20.0 \mathrm {~cm}$

Accepted Answer

The answer of Solve the problem.
-A pendulum bob swings \(5.0...

Question 11

Solve the problem.
-A sequence of yearly payments of $3000 is invested at an interest rate of 2%, compounded annually. What is the total amount of the annuity after 8 years?
A) $26,748.91
B) $25,748.91
C) $25,248.91
D) $24,748.91

Accepted Answer

The answer of Solve the problem.
-A sequence of yearly payments...

Question 12

Find the common ratio r for the given infinite geometric sequence.
-$3,9,27,81,243 , \ldots$
A) $\frac { 1 } { 3 }$
B) 3
C) $- 3$
D) 9

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 (9) by the first term (3) gives $r = 9 / 3 = 3$.

Question 13

Solve the problem.
-A ball is dropped from a height of $5.0 \mathrm {~m}$. On each upward bounce the ball returns to $\frac { 2 } { 5 }$ of its previous height. Find the total vertical distance the ball travels before coming to rest.
A) $12.9 \mathrm {~m}$
B) $12.6 \mathrm {~m}$
C) $8.3 \mathrm {~m}$
D) $11.6 \mathrm {~m}$

Accepted Answer

The answer of Solve the problem.
-A ball is dropped from...

Question 14

Solve the problem.
-A sequence of yearly payments of $1500 is invested at the end of every compounding period, where interest is compounded semi-annually at 1%. What is the total amount of the annuity at the
End of 4 years?
A) $162,212.11
B) $24,857.01
C) $12,212.11
D) $168,318.17

Accepted Answer

The answer of Solve the problem.
-A sequence of yearly payments...

Question 15

Solve the problem.
-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) 14,641 people
B) 6105 people
C) 15,000 people
D) 16,105 people

Accepted Answer

The population increases by 10% each year, which can be calculated using the formula for compound interest: P = P0(1 + r)^t, where P0 is the initial population, r is the rate of increase, and t is the time in years. Here, P0 = 10,000, r = 0.10, and t = 5. So, P = 10,000(1 + 0.10)^5 = 10,000(1.10)^5 = 16,105.1, which rounds to 16,105 people.

Question 16

Evaluate the series, if it converges.
-$$\sum _ { i = 1 } ^ { \infty } 3 \left( - \frac { 9 } { 4 } \right) ^ { i - 1 }$$
A) $\frac { 12 } { 13 }$
B) Does not converge.
C) $- \frac { 27 } { 13 }$
D) $\frac { 4 } { 13 }$

Accepted Answer

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

Question 17

Find the common ratio r for the given infinite geometric sequence.
-80, 40, 20, 10, 5, . . .
A) 0.5
B) 2
C) 5
D) -0.5

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 (40) by the first term (80) gives $r = \frac{40}{80} = 0.5$.

Question 18

Find the common ratio r for the given infinite geometric sequence.
-$\frac { 3 } { 2 } , \frac { 3 } { 4 } , \frac { 3 } { 8 } , \frac { 3 } { 16 } , \frac { 3 } { 32 } , \ldots$
A) 2
B) $\frac { 1 } { 2 }$
C) $\frac { 1 } { 6 }$
D) 6

Accepted Answer

The common ratio $r$ can be found by dividing any term in the sequence by the preceding term. For instance, $\frac{3/4}{3/2} = \frac{1}{2}$, indicating that the common ratio $r$ is $\frac{1}{2}$.

Question 19

Evaluate the series, if it converges.
-$$\sum _ { i = 1 } ^ { \infty } 46 \left( - \frac { 5 } { 7 } \right) ^ { i - 1 }$$
A) Does not converge.
B) $- \frac { 115 } { 6 }$
C) $\frac { 7 } { 12 }$
D) $\frac { 161 } { 6 }$

Accepted Answer

The series is a geometric series with first term $a = 46$ and common ratio $r = -\frac{5}{7}$. Since $|r| < 1$, the series converges to $\frac{a}{1-r} = \frac{46}{1 - (-\frac{5}{7})} = \frac{46}{1 + \frac{5}{7}} = \frac{46}{\frac{12}{7}} = \frac{46 \times 7}{12} = \frac{322}{12} = \frac{161}{6}$.

Question 20

Find the common ratio r for the given infinite geometric sequence.
-$\frac { 4 } { 3 } , \frac { 16 } { 3 } , \frac { 64 } { 3 } , \frac { 256 } { 3 } , \frac { 1024 } { 3 } , \ldots$
A) 12
B) $\frac { 1 } { 4 }$
C) 4
D) 10

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 $\frac{16}{3}$ by the first term $\frac{4}{3}$ gives $r = 4$.

Quiz 11: Further Topics in Algebra

Solve the problem. -A town has a population of 1000 people and is increasing by 9% every year. What will the population be at the end of 9 years?

Solve the problem. -A man borrowed $2000 at 2.5% interest compounded annually. If he paid off the loan in full at the end of 6 years, how much did he pay?

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

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

Use a graphing calculator to evaluate the sum. Round to the nearest thousandth. - $\sum _ { i = 2 } ^ { 9 } - 2 ( 0.93 ) i$

Evaluate the series, if it converges. - $\sum _ { i = 1 } ^ { \infty } 25 \left( \frac { 7 } { 10 } \right) ^ { i - 1 }$

Solve the problem. -A woman borrowed $10,000 at 2% interest compounded annually. If the loan was paid off in one lump sum at the end of 5 years, how much did she pay?

Use a graphing calculator to evaluate the sum. Round to the nearest thousandth. - $\sum _ { \mathrm { k } = 1 } ^ { 6 } ( 1.41 ) ^ { \mathrm { k } }$

Evaluate the series, if it converges. - $\sum _ { i = 1 } ^ { \infty } 24 \left( \frac { 7 } { 2 } \right) ^ { i - 1 }$

Solve the problem. -A pendulum bob swings $5.0 \mathrm {~cm}$ on its first oscillation. On each subsequent oscillation the bob travels $\frac { 3 } { 5 }$ of the previous distance. Find the total distance the bob travels before coming to rest.

Solve the problem. -A sequence of yearly payments of $3000 is invested at an interest rate of 2%, compounded annually. What is the total amount of the annuity after 8 years?

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

Solve the problem. -A ball is dropped from a height of $5.0 \mathrm {~m}$ . On each upward bounce the ball returns to $\frac { 2 } { 5 }$ of its previous height. Find the total vertical distance the ball travels before coming to rest.

Solve the problem. -A sequence of yearly payments of $1500 is invested at the end of every compounding period, where interest is compounded semi-annually at 1%. What is the total amount of the annuity at the End of 4 years?

Solve the problem. -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?

Evaluate the series, if it converges. - $\sum _ { i = 1 } ^ { \infty } 3 \left( - \frac { 9 } { 4 } \right) ^ { i - 1 }$

Find the common ratio r for the given infinite geometric sequence. -80, 40, 20, 10, 5, . . .

Find the common ratio r for the given infinite geometric sequence. - $\frac { 3 } { 2 } , \frac { 3 } { 4 } , \frac { 3 } { 8 } , \frac { 3 } { 16 } , \frac { 3 } { 32 } , \ldots$

Evaluate the series, if it converges. - $\sum _ { i = 1 } ^ { \infty } 46 \left( - \frac { 5 } { 7 } \right) ^ { i - 1 }$

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 11: Further Topics in Algebra

Solve the problem. -A town has a population of 1000 people and is increasing by 9% every year. What will the population be at the end of 9 years?

Solve the problem. -A man borrowed $2000 at 2.5% interest compounded annually. If he paid off the loan in full at the end of 6 years, how much did he pay?

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

Find the common ratio r for the given infinite geometric sequence. - 3,−9,27,−81,243,…3 , - 9,27 , - 81,243 , \ldots3,−9,27,−81,243,…

Use a graphing calculator to evaluate the sum. Round to the nearest thousandth. - ∑i=29−2(0.93) i\sum _ { i = 2 } ^ { 9 } - 2 ( 0.93 ) i∑i=29​−2(0.93) i

Evaluate the series, if it converges. - ∑i=1∞25(710) i−1\sum _ { i = 1 } ^ { \infty } 25 \left( \frac { 7 } { 10 } \right) ^ { i - 1 }∑i=1∞​25(107​) i−1

Solve the problem. -A woman borrowed $10,000 at 2% interest compounded annually. If the loan was paid off in one lump sum at the end of 5 years, how much did she pay?

Use a graphing calculator to evaluate the sum. Round to the nearest thousandth. - ∑k=16(1.41) k\sum _ { \mathrm { k } = 1 } ^ { 6 } ( 1.41 ) ^ { \mathrm { k } }∑k=16​(1.41) k

Evaluate the series, if it converges. - ∑i=1∞24(72) i−1\sum _ { i = 1 } ^ { \infty } 24 \left( \frac { 7 } { 2 } \right) ^ { i - 1 }∑i=1∞​24(27​) i−1

Solve the problem. -A pendulum bob swings 5.0 cm5.0 \mathrm {~cm}5.0 cm on its first oscillation. On each subsequent oscillation the bob travels 35\frac { 3 } { 5 }53​ of the previous distance. Find the total distance the bob travels before coming to rest.