Geometric Sequences Practice

Question

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Question
Jamie and Eddie are buying an apartment for $320,000. Their loan officer tells them that their principal will decrease by 4.5% every year if they make the minimum mortgage payments. Complete the table to predict the principal for the first 3 years. (Round to the nearest hundredth, if applicable.)(2 points)
Year Principal
0 $320,000.00
1 $
2 $
3 $
Remaining Attempts : 3
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Answer 1

To calculate the principal for each year based on a 4.5% decrease, we can use the formula for a geometric sequence:

\[ P_n = P_0 \times (1 - r)^n \]

where:

• \( P_n \) is the principal after \( n \) years,
• \( P_0 \) is the initial principal,
• \( r \) is the rate of decrease (as a decimal),
• \( n \) is the number of years.

For this problem, the values are:

• \( P_0 = 320,000 \)
• \( r = 0.045 \)

Year 1:

\[ P_1 = 320,000 \times (1 - 0.045)^1 = 320,000 \times 0.955 = 305,600 \]

Year 2:

\[ P_2 = 320,000 \times (1 - 0.045)^2 = 320,000 \times (0.955)^2 \] Calculating \( (0.955)^2 \): \[ 0.955^2 = 0.912025 \quad \text{(approx)} \] Now, calculate \( P_2 \): \[ P_2 = 320,000 \times 0.912025 \approx 291,268 \]

Year 3:

\[ P_3 = 320,000 \times (1 - 0.045)^3 = 320,000 \times (0.955)^3 \] Calculating \( (0.955)^3 \): \[ 0.955^3 \approx 0.870393 \quad \text{(approx)} \] Now, calculate \( P_3 \): \[ P_3 = 320,000 \times 0.870393 \approx 278,125 \]

Summary of the results:

• Year 1: $305,600.00
• Year 2: $291,268.00
• Year 3: $278,125.00

So, the table will be filled out as:

• Year 1: $305,600.00
• Year 2: $291,268.00
• Year 3: $278,125.00