Question

Aria buys a new car for $36,000. She learns that every year her car will depreciate in value by 15%. This means that each year her car will be worth 85% of what it was the previous year. Aria tracks how much her car is worth over time.

Create a sequence that describes the worth of Aria’s car each year for 8 years. (Write answers in the form or 2,000 or 1,567.87).

Create a formula for the worth of Aria’s car over time.

Use your formula to find how much Aria’s car is worth in 20 years

Answer 1

To create the sequence that describes the worth of Aria's car each year for 8 years, we first need to recognize that the car's value depreciates by 15% each year. This means that after one year, the car is worth 85% of its previous value.

The formula to describe the worth of the car after \( n \) years can be written as:
\[
V(n) = V_0 \times (0.85)^n
\]
where:
- \( V_0 \) is the initial value of the car ($36,000),
- \( n \) is the number of years,
- \( V(n) \) is the value of the car after \( n \) years.

Now we can generate the sequence for the worth of Aria’s car for the first 8 years:

1. **Year 0**:
\[
V(0) = 36000 \times (0.85)^0 = 36000
\]

2. **Year 1**:
\[
V(1) = 36000 \times (0.85)^1 = 36000 \times 0.85 = 30600.00
\]

3. **Year 2**:
\[
V(2) = 36000 \times (0.85)^2 = 36000 \times 0.7225 = 26010.00
\]

4. **Year 3**:
\[
V(3) = 36000 \times (0.85)^3 = 36000 \times 0.614125 = 22093.50
\]

5. **Year 4**:
\[
V(4) = 36000 \times (0.85)^4 = 36000 \times 0.52200625 = 18872.23
\]

6. **Year 5**:
\[
V(5) = 36000 \times (0.85)^5 = 36000 \times 0.4437053125 = 15973.40
\]

7. **Year 6**:
\[
V(6) = 36000 \times (0.85)^6 = 36000 \times 0.3768890578125 = 13587.01
\]

8. **Year 7**:
\[
V(7) = 36000 \times (0.85)^7 = 36000 \times 0.320507703125 = 11579.78
\]

9. **Year 8**:
\[
V(8) = 36000 \times (0.85)^8 = 36000 \times 0.2724907995625 = 9809.87
\]

### Summary of the Sequence
- Year 0: $36,000.00
- Year 1: $30,600.00
- Year 2: $26,010.00
- Year 3: $22,093.50
- Year 4: $18,872.23
- Year 5: $15,973.40
- Year 6: $13,587.01
- Year 7: $11,579.78
- Year 8: $9,809.87

### Formula for the Worth of Aria's Car
The formula we've created is:
\[
V(n) = 36000 \times (0.85)^n
\]

### Value of Aria's Car in 20 Years
To find the worth of the car in 20 years, we substitute \( n = 20 \) into the formula:

\[
V(20) = 36000 \times (0.85)^{20}
\]

Calculating \( (0.85)^{20} \):

\[
(0.85)^{20} \approx 0.038759 \quad (\text{using a calculator})
\]

Then:

\[
V(20) \approx 36000 \times 0.038759 \approx 1395.32
\]

### Final Result
Thus, the worth of Aria's car in 20 years will be approximately **$1,395.32**.