Question 1

Find the future value of $16,000 invested for 15 years at 9%, compounded daily. Round your answer to the nearest cent.

Accepted Answer

The answer of Find the future value of $16,000 invested...

Question 2

If $3600 is invested at 8% interest, compounded annually, how long will it take the account to reach $9790.65?

Accepted Answer

The answer of If $3600 is invested at 8% interest,...

Question 3

Find the sum of the first 6 terms of the arithmetic sequence 2, 4, 6, . . . .

Accepted Answer

The sum of the first 6 terms of an arithmetic sequence can be found using the formula $S_n = \frac{n}{2}(a_1 + a_n)$, where $S_n$ is the sum of the first $n$ terms, $a_1$ is the first term, and $a_n$ is the $n$th term. Here, $n = 6$, $a_1 = 2$, and $a_n = 2 + 5(2) = 12$ (since the common difference $d = 2$). Thus, $S_6 = \frac{6}{2}(2 + 12) = 3(14) = 42$.

Question 4

The population of a town was 27,600 when the first official population was recorded. If the population decreased 350 people per year, how many people lived in the town 15 years later?

Accepted Answer

The population decreases by 350 people per year. Over 15 years, the decrease is 350 * 15 = 5,250 people. Subtracting this from the initial population: 27,600 - 5,250 = 22,350.

Question 5

Find the sum of the first 4 terms of the arithmetic sequence with first term  -13  and common difference  -2 .

Accepted Answer

The sum of the first 4 terms of an arithmetic sequence can be found using the formula $S_n = \frac{n}{2} [2a + (n-1)d]$, where $S_n$ is the sum of the first $n$ terms, $a$ is the first term, $d$ is the common difference, and $n$ is the number of terms. Plugging in the values given ($a = -13$, $d = -2$, $n = 4$), we get $S_4 = \frac{4}{2} [2(-13) + (4-1)(-2)] = 2[-26 - 6] = 2[-32] = -64$.

Question 6

Find the sum of the first 5 terms of the geometric sequence  1,-4,16, ...

Accepted Answer

The common ratio (r) of the sequence is -4 (each term is -4 times the previous one). The sum of the first n terms of a geometric sequence is given by S_n = a(1 - r^n) / (1 - r), where a is the first term and r is the common ratio. Plugging in the values: S_5 = 1(1 - (-4)^5) / (1 - (-4)) = 1(1 + 1024) / 5 = 1025 / 5 = 205.

Question 7

Find the sum of the first 5 terms of the geometric sequence 4, 12, 36, . . . .

Accepted Answer

The sequence is a geometric sequence where each term is multiplied by 3 to get the next term (common ratio = 3). The first term (a) is 4. The sum of the first n terms of a geometric sequence is given by S_n = a(1 - r^n) / (1 - r), where r is the common ratio. For the first 5 terms, S_5 = 4(1 - 3^5) / (1 - 3) = 4(1 - 243) / (-2) = 4 * 242 / 2 = 484.

Question 8

Find the sum of the first 10 terms of the arithmetic sequence with first term 11 and common difference  -4 .

Accepted Answer

The answer of Find the sum of the first 10...

Question 9

A new firm loses $3600 in its first month, but its profit increases by $700 in each succeeding month. What is its profit in the 8th month?

Accepted Answer

The firm's profit increases by $700 each month. In the 8th month, the profit would be calculated as follows: Initial loss = -$3600, so for the first month, it's -$3600. From the second month onwards, it starts to make a profit increase of $700 per month. So, by the 8th month, the calculation is -$3600 + 7($700) = -$3600 + $4900 = $1300.

Question 10

Find the sum of the first 4 terms of the arithmetic sequence with first term 1 and common difference  -2 .

Accepted Answer

The first four terms of the sequence are 1, -1, -3, and -5. Their sum is 1 + (-1) + (-3) + (-5) = -8.

Question 11

Eloise contracts to work for 10 days, receiving $0.03 the first day, $0.09 the second day, $0.27 the third day, and so on, with each day's pay triple that of the previous day. How much will she earn on the last day of the
Contract?

Accepted Answer

Eloise's earnings follow a geometric sequence where each term is triple the previous one. The first day's pay is $0.03, and the ratio (r) is 3. The formula for the nth term of a geometric sequence is $a_n = a_1 \cdot r^{(n-1)}$. For the 10th day, $a_{10} = 0.03 \cdot 3^{(10-1)} = 0.03 \cdot 3^9 = 0.03 \cdot 19683 = $590.49$.

Question 12

Find the sum of the first 11 terms of the arithmetic sequence 1, 3, 5, . . . .

Accepted Answer

The sequence is arithmetic with the first term a = 1 and common difference d = 2. The sum of the first n terms of an arithmetic sequence is given by S_n = n/2 * (2a + (n-1)d). For the first 11 terms, S_11 = 11/2 * (2*1 + (11-1)*2) = 11/2 * (2 + 20) = 11/2 * 22 = 121.

Question 13

Find the sum of the first 5 terms of the geometric sequence $\frac { 2 } { 3 } , \frac { 4 } { 3 } , \frac { 8 } { 3 } , \ldots$

Accepted Answer

The answer of Find the sum of the first 5...

Question 14

A man borrowed $3000 at 10% interest compounded annually. If he paid off the loan in full at the end of 4 years, how much did he pay? Round your answer to the nearest cent.

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 ($3000), r is the annual interest rate (10% or 0.10), and n is the number of years (4). Plugging these values in gives A = 3000(1 + 0.10)^4 = $4392.30.

Question 15

A certain ball when dropped from a height rebounds 107 of the original height. How high will the ball rebound after the fifth bounce if it was dropped from a height of 32 feet? Round your answer to the nearest tenth of a
Foot.

Accepted Answer

The answer of A certain ball when dropped from a...

Question 16

Find the sum of the first 10 terms of the arithmetic sequence 88, 159, 230, . . . .

Accepted Answer

The common difference (d) in the sequence is 159 - 88 = 71. The first term (a) is 88. The sum of the first n terms of an arithmetic sequence is given by $S_n = \frac{n}{2}[2a + (n-1)d]$. Substituting n = 10, a = 88, and d = 71, we get $S_{10} = \frac{10}{2}[2(88) + (10-1)71] = 5[176 + 9(71)] = 5[176 + 639] = 5[815] = 4075$.

Question 17

Find the sum of the first 5 terms of the geometric sequence with first term 1 and common ratio  -3 .

Accepted Answer

The answer of Find the sum of the first 5...

Question 18

Write four additional terms of the geometric sequence.
-A man earned $3000 the first year he worked. If he received a raise of $400 at the end of each year, what was his salary during the 15th year?

Accepted Answer

The salary increases by $400 each year, so the salary in the 15th year is $3000 + 14*$400 = $8600.

Question 19

Suppose a plumber charges $160 for the first hour plus $70 for each additional hour. Write an expression for the cost (in dollars) of a job lasting n hours.

Accepted Answer

The correct expression is $160 + 70 ( n - 1 )$, because the first hour costs $160 and each additional hour (n-1 hours) costs $70.

Question 20

A woman borrowed $30,000 at 8% interest compounded annually. If the loan was paid off at the end of 5 years, how much did she pay? Round your answer to the nearest cent.

Accepted Answer

The amount paid off at the end of 5 years 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 (8% or 0.08), and n is the number of years (5). Plugging in the values: A = $30,000(1 + 0.08)^5 = $44,079.84.

Quiz 8: Special Topics in Algebra

Find the future value of $16,000 invested for 15 years at 9%, compounded daily. Round your answer to the nearest cent.

If $3600 is invested at 8% interest, compounded annually, how long will it take the account to reach $9790.65?

Find the sum of the first 6 terms of the arithmetic sequence 2, 4, 6, . . . .

The population of a town was 27,600 when the first official population was recorded. If the population decreased 350 people per year, how many people lived in the town 15 years later?

Find the sum of the first 4 terms of the arithmetic sequence with first term -13 and common difference -2 .

Find the sum of the first 5 terms of the geometric sequence 1,-4,16, ...

Find the sum of the first 5 terms of the geometric sequence 4, 12, 36, . . . .

Find the sum of the first 10 terms of the arithmetic sequence with first term 11 and common difference -4 .

A new firm loses $3600 in its first month, but its profit increases by $700 in each succeeding month. What is its profit in the 8th month?

Find the sum of the first 4 terms of the arithmetic sequence with first term 1 and common difference -2 .

Eloise contracts to work for 10 days, receiving $0.03 the first day, $0.09 the second day, $0.27 the third day, and so on, with each day's pay triple that of the previous day. How much will she earn on the last day of the Contract?

Find the sum of the first 11 terms of the arithmetic sequence 1, 3, 5, . . . .

Find the sum of the first 5 terms of the geometric sequence 23,43,83,…\frac { 2 } { 3 } , \frac { 4 } { 3 } , \frac { 8 } { 3 } , \ldots32​,34​,38​,…

A man borrowed $3000 at 10% interest compounded annually. If he paid off the loan in full at the end of 4 years, how much did he pay? Round your answer to the nearest cent.

A certain ball when dropped from a height rebounds 107 of the original height. How high will the ball rebound after the fifth bounce if it was dropped from a height of 32 feet? Round your answer to the nearest tenth of a Foot.

Find the sum of the first 10 terms of the arithmetic sequence 88, 159, 230, . . . .

Find the sum of the first 5 terms of the geometric sequence with first term 1 and common ratio -3 .

Write four additional terms of the geometric sequence. -A man earned $3000 the first year he worked. If he received a raise of $400 at the end of each year, what was his salary during the 15th year?

Suppose a plumber charges $160 for the first hour plus $70 for each additional hour. Write an expression for the cost (in dollars) of a job lasting n hours.

A woman borrowed $30,000 at 8% interest compounded annually. If the loan was paid off at the end of 5 years, how much did she pay? Round your answer to the nearest cent.

Find the sum of the first 5 terms of the geometric sequence $\frac { 2 } { 3 } , \frac { 4 } { 3 } , \frac { 8 } { 3 } , \ldots$