You buy a new car for 24,000 . At the end of n years, the...

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

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

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Key Concepts

Geometric Sequence

A geometric sequence is a sequence of numbers where each term is derived by multiplying the previous term by a constant factor known as the common ratio. This structure allows for the concise formulation of the nth term and is widely used in modeling processes that involve exponential growth or decay.

Exponential Decay

Exponential decay refers to a decrease at a rate proportional to the current value, leading to a constant percentage decrease in each time period. This principle is mathematically represented by multiplying the initial value by a constant factor raised to the power of the number of time periods elapsed, and it is commonly used to model depreciation, radioactive decay, and similar processes.

Depreciation

Depreciation is the process by which an asset loses value over time, often modeled using exponential decay. By applying a constant percentage decrease each period, the asset’s value can be expressed as a geometric sequence where the initial cost is reduced consistently, reflecting the diminishing value with each passing year.

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