Problem 80 - College Algebra

Question

You buy a new car for \(\$ 24,000 .\) At the end of \(n\) years, the value of your car is given by the sequence $$a_{n}-24,000\left(\frac{3}{4}\right)^{n}, \quad n-1,2,3, \dots$$ Find \(a_{5}\) and write a sentence explaining what this value represents. Describe the \(n\) the term of the sequence in terms of the value of your car at the end of each year.

Step-by-Step Solution

Answer

The value of the car at the end of the 5th year, represented by \(a_5\), is approximately \$5,062.50. This value represents how much the car has depreciated after five years of use. The nth term of the sequence represents the value of the car at the end of n years, assuming a yearly depreciation of 25%.

Step 1: Identify the formula and the parameters

The depreciation of the car's value is modelled by the sequence \(a_n = 24,000 * (3/4)^n\), with n = 1, 2, 3, .... This formula is applied for each year that passes, starting from n=1 (the end of the first year).

Step 2: Apply the formula for the 5th year

To find the value of the car at the end of the 5th year (\(a_5\)), substitute n with 5 in the formula: \(a_5 = 24,000 * (3/4)^5\). Calculating, we find that \(a_5 = 5,062.50\) dollars.

Step 3: Interpret the obtained result

\(a_5 = 5,062.50\) dollars represents the value of the car at the end of the 5th year. This is showing how much the car has depreciated after five years.

Step 4: Describe the nth term of the sequence in terms of the car value.

The nth term of the sequence, \(a_n = 24,000 * (3/4)^n\), represents the value of the car at the end of the nth year. With each year that passes, the value of your car depreciates by 25%, hence the factor of (\(3/4)^n\). The depreciation starts from the initial value of \$24,000 and gets compounded year by year.

Key Concepts

Depreciation

Depreciation refers to the reduction in value of an asset over time. For cars, this is a common phenomenon due to wear and tear or advancements in technology. When we talk about car depreciation in a sequence, specifically, it shows how much the car’s value will decrease each year.
The given formula, \( a_n = 24,000 \times \left( \frac{3}{4} \right)^n \), reflects how the car’s worth changes yearly. The factor \( \frac{3}{4} \) indicates that the car retains 75% of its value each year and depreciates by 25%. This is a geometric progression since the car’s value decreases proportionally with each passing year.

Nth Term

The nth term in a sequence formula gives you a specific value that corresponds to a certain position within the sequence. In this context, "n" represents the number of years. For a car value sequence, the nth term tells you the car's value at the end of "n" years.
You can determine any year's value using \( a_n = 24,000 \times \left( \frac{3}{4} \right)^n \). For example, \( a_5 \) means the car's value at the end of year five. Knowing the nth term helps you predict future values and understand how depreciation affects the car over time.

Sequence Formula

A sequence formula provides a systematic method to calculate the terms of a sequence. Here, the formula \( a_n = 24,000 \times \left( \frac{3}{4} \right)^n \) is used to model how the car's value decreases each year.

• The initial term is \( 24,000 \), representing the purchase price of the car.
• The common ratio is \( \frac{3}{4} \), which reflects a 25% decrease annually.
• Using this formula helps determine the car's value after any number of years \( n \).

This is essential for planning finances, knowing when the car’s value might reach a lower threshold, or deciding when to sell or trade in your vehicle.

Car Value

Understanding car value in terms of depreciation helps you make informed decisions. Initially, the car is worth \( \$24,000 \). With each passing year, this value declines as described by the sequence formula.
For instance, after five years, the car is valued at \( 5,062.50 \) dollars, indicating significant depreciation. Recognizing the changing car value is helpful for:

• Resale and trade-in strategies.
• Setting budgets for maintenance and insurance based on the car’s worth.
• Forecasting long-term financial impact due to depreciation.

By tracking these changes, you can better manage your asset over time.