Question 1(Multiple Choice Worth 4 points)
(Compound Interest and Geometric Sequences LC)

A geometric sequence begins with −4, 20, −100, 500, …

Which option below represents the formula for the sequence?

f(n) = −5(−4)n−1
f(n) = −4(−5)n−1
f(n) = −5(−4)n
f(n) = −4(−5)n
Question 2(Multiple Choice Worth 4 points)
(Continuously Compounded Interest MC)

A college student plans to use a credit card to cover the cost of purchasing a $3,200 campus meal plan. If the credit card has annual interest rate of 12.99% compounded continuously and the student plans to pay off the loan in 15 months, how much interest will be owed?

$739.25
$688.41
$588.69
$564.16

### Question 1: Geometric Sequence Formula

Given the geometric sequence: $-4, 20, -100, 500, \ldots$

To find the common ratio, $ r $:
\[
r = \frac{20}{-4} = -5 \\
r = \frac{-100}{20} = -5 \\
r = \frac{500}{-100} = -5
\]
The common ratio $ r $ is $-5$.

The general formula for a geometric sequence is:
\[
f(n) = a \cdot r^{(n-1)}
\]
where $ a $ is the first term, and $ r $ is the common ratio.

Here, $ a = -4 $ and $ r = -5 $. Thus, the formula for the sequence is:
\[
f(n) = -4 \cdot (-5)^{(n-1)}
\]

Therefore, the correct option is:
\[ \boxed{f(n) = -4(-5)^{n-1}} \]

### Question 2: Continuously Compounded Interest

Given:
- Principal amount $ P = \$3200 $
- Annual interest rate $ r = 12.99\% = 0.1299 $
- Time $ t = 15 \text{ months} = \frac{15}{12} \text{ years} = 1.25 \text{ years} $

The formula for continuously compounded interest is:
\[
A = P \cdot e^{rt}
\]
where $ A $ is the amount after time $ t $.

Substitute the given values into the formula:
\[
A = 3200 \cdot e^{0.1299 \cdot 1.25}
\]

First, calculate the exponent:
\[
0.1299 \times 1.25 = 0.162375
\]

Next, calculate $ e^{0.162375} $:
\[
e^{0.162375} \approx 1.1767
\]

Now, calculate $ A $:
\[
A = 3200 \times 1.1767 \approx 3765.44
\]

The interest owed is the difference between the amount $ A $ and the principal $ P $:
\[
\text{Interest} = 3765.44 - 3200 = 565.44
\]

Rounding to the nearest cent, the closest option is:
\[ \boxed{564.16} \]

### 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 determine the common ratio in a geometric sequence?
2. What is the general formula for a geometric sequence?
3. How does continuous compounding differ from regular compounding?
4. What is the value of $ e $ in continuously compounded interest calculations?
5. Can you explain how to convert months to years for interest calculations?

### Tip:
For geometric sequences, always verify the common ratio by dividing each term by its preceding term to ensure consistency.

Explore solutions to geometric sequence problems and continuous compounded interest calculations. Learn the formula for geometric sequences and how to calculate interest compounded continuously. Practice with examples and step-by-step solutions.

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