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:
-
Year 0:
\[ V(0) = 36000 \times (0.85)^0 = 36000 \] -
Year 1:
\[ V(1) = 36000 \times (0.85)^1 = 36000 \times 0.85 = 30600.00 \] -
Year 2:
\[ V(2) = 36000 \times (0.85)^2 = 36000 \times 0.7225 = 26010.00 \] -
Year 3:
\[ V(3) = 36000 \times (0.85)^3 = 36000 \times 0.614125 = 22093.50 \] -
Year 4:
\[ V(4) = 36000 \times (0.85)^4 = 36000 \times 0.52200625 = 18872.23 \] -
Year 5:
\[ V(5) = 36000 \times (0.85)^5 = 36000 \times 0.4437053125 = 15973.40 \] -
Year 6:
\[ V(6) = 36000 \times (0.85)^6 = 36000 \times 0.3768890578125 = 13587.01 \] -
Year 7:
\[ V(7) = 36000 \times (0.85)^7 = 36000 \times 0.320507703125 = 11579.78 \] -
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.