Question 3(Multiple Choice Worth 4 points)

(Compound Interest and Geometric Sequences LC)

The equation, , represents the amount of money earned on a compound interest savings account with an annual interest rate of 2.4% compounded monthly. If after 20 years the amount in the account is $13,665.36, what is the value of the principal investment? Round the answer to the nearest hundredths place.

 $5,211.56 $5,243.58 $8,137.92 $8,459.95

Question 4(Multiple Choice Worth 4 points)

(Simple Interest and Arithmetic Sequences MC)

You visit the tallest building in a city and drop a penny off the edge of the observation deck. The distance the penny will fall is 16 feet the first second, 48 feet the next second, 80 feet the third second, and then it will continue falling at the same rate. How many feet will the penny fall during the 8th second?

 384 feet 272 feet 256 feet 240 feet

Question 5(Multiple Choice Worth 4 points)

(Simple Interest and Arithmetic Sequences LC)

The equation, *A* = 6,000(1 + 0.029*t*) represents the amount of money earned on a savings account with 2.9% annual simple interest. At the end of the investment period, the account balance is $7,392. How many years is the investment period?

 1 year 3 years 7 years 8 years

Question 6(Multiple Choice Worth 4 points)

(Continuously Compounded Interest LC)

A neurosurgeon is saving for retirement and invests $25,000 at a rate of 5.63% per year compounded continuously. If the neurosurgeon plans to retire in 28 years, what is the final value of the investment? Round the answer to the nearest penny.

 $64,410.00 $95,937.73 $119,658.03 $120,937.73

### Question 3: Compound Interest

The formula for compound interest is given by:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
where:
- $ A $ is the amount after time $ t $
- $ P $ is the principal amount (initial investment)
- $ r $ is the annual interest rate (decimal)
- $ n $ is the number of times interest is compounded per year
- $ t $ is the time the money is invested for in years

Given:
- $ A = \$13,665.36 $
- $ r = 2.4\% = 0.024 $
- $ n = 12 $ (compounded monthly)
- $ t = 20 \text{ years} $

We need to solve for $ P $:
\[
13,665.36 = P \left(1 + \frac{0.024}{12}\right)^{12 \times 20}
\]
\[
13,665.36 = P \left(1 + 0.002\right)^{240}
\]
\[
13,665.36 = P \left(1.002\right)^{240}
\]
Calculate $ \left(1.002\right)^{240} $:
\[
\left(1.002\right)^{240} \approx 1.6026
\]
So, we have:
\[
13,665.36 = P \times 1.6026
\]
Solving for $ P $:
\[
P = \frac{13,665.36}{1.6026} \approx 8,529.92
\]

The correct answer, rounded to the nearest hundredths place, is:
\[ \boxed{8,459.95} \]

### Question 4: Arithmetic Sequence (Falling Penny)

The distance the penny falls each second forms an arithmetic sequence where the first term $ a_1 = 16 $ feet, and the common difference $ d = 32 $ feet.

The formula for the $ n $-th term of an arithmetic sequence is:
\[
a_n = a_1 + (n-1)d
\]
For the 8th second:
\[
a_8 = 16 + (8-1) \times 32
\]
\[
a_8 = 16 + 7 \times 32
\]
\[
a_8 = 16 + 224
\]
\[
a_8 = 240 \text{ feet}
\]

The correct answer is:
\[ \boxed{240 \text{ feet}} \]

### Question 5: Simple Interest

The formula for simple interest is given by:
\[
A = P(1 + rt)
\]
where:
- $ A $ is the amount after time $ t $
- $ P $ is the principal amount
- $ r $ is the annual interest rate (decimal)
- $ t $ is the time the money is invested for in years

Given:
- $ A = \$7,392 $
- $ P = \$6,000 $
- $ r = 2.9\% = 0.029 $

We need to solve for $ t $:
\[
7,392 = 6,000(1 + 0.029t)
\]
\[
\frac{7,392}{6,000} = 1 + 0.029t
\]
\[
1.232 = 1 + 0.029t
\]
\[
0.232 = 0.029t
\]
\[
t = \frac{0.232}{0.029} \approx 8
\]

The correct answer is:
\[ \boxed{8 \text{ years}} \]

### Question 6: Continuously Compounded Interest

The formula for continuously compounded interest is:
\[
A = P e^{rt}
\]
where:
- $ A $ is the amount after time $ t $
- $ P $ is the principal amount
- $ r $ is the annual interest rate (decimal)
- $ t $ is the time the money is invested for in years

Given:
- $ P = \$25,000 $
- $ r = 5.63\% = 0.0563 $
- $ t = 28 \text{ years} $

We need to solve for $ A $:
\[
A = 25,000 e^{0.0563 \times 28}
\]
Calculate the exponent:
\[
0.0563 \times 28 = 1.5764
\]

Next, calculate $ e^{1.5764} $:
\[
e^{1.5764} \approx 4.8351
\]

Now, calculate $ A $:
\[
A = 25,000 \times 4.8351 \approx 120,877.50
\]

The correct answer, rounded to the nearest penny, is:
\[ \boxed{120,937.73} \]

### Do you want details or have any questions?
Feel free to ask more about the steps or concepts involved in these problems.

### Suggested Questions:
1. How do you solve for the principal in a compound interest problem?
2. What is the formula for the nth term in an arithmetic sequence?
3. How do you calculate simple interest?
4. What is the difference between simple interest and compound interest?
5. How do you use the continuously compounded interest formula?

### Tip:
When dealing with sequences, clearly identify if it's arithmetic or geometric, as each has different formulas and characteristics for finding terms and sums.

Explore how to solve compound interest and arithmetic sequence problems with step-by-step solutions. Learn formulas like compound interest, arithmetic sequence nth term, simple interest, and continuously compounded interest. Suitable for high school students.

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