Solve each application. A luxury car purchased for 64,000...

Solve each application. A luxury car purchased for $\$ 64,000$ depreciates $15 \%$ each year. a) Find the general term, $a_{n}$, of the geometric sequence that models the value of the car at the beginning of each year. b) How much is the luxury car worth at the beginning of the fourth year?

Solve each application.
A luxury car purchased for $\$ 64,000$ depreciates $15 \%$ each year.
a) Find the general term, $a_{n}$, of the geometric sequence that models the value of the car at the beginning of each year.
b) How much is the luxury car worth at the beginning of the fourth year?

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Key Concepts

Exponential Decay

Exponential decay describes a process in which a quantity decreases at a rate proportional to its current value. This concept is crucial in real-world applications like depreciation, where an asset's value diminishes continuously by a constant percentage over time.

Common Ratio

The common ratio is the constant multiplier used to move from one term to the next in a geometric sequence. When modeling depreciation, the common ratio is determined by subtracting the depreciation rate from 1, which reflects the proportion of the asset's value that remains after each time period.

Geometric Sequence

A geometric sequence is a series of numbers where each term is found by multiplying the previous term by a constant factor known as the common ratio. This concept is fundamental for modeling situations with consistent percentage changes over time, such as depreciation or growth scenarios.

Initial Value

The initial value is the starting point or first term in a sequence or function. In depreciation problems, this represents the original cost or value of an asset before any decrease occurs, providing a baseline for all subsequent calculations.

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