Math Problem Statement
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
Solution
Question 1: Geometric Sequence Formula
Given the geometric sequence:
To find the common ratio, :
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.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Geometric Sequences
Compound Interest
Continuous Compounding
Formulas
Geometric sequence formula
Continuous compound interest formula
Theorems
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Suitable Grade Level
High School
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