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

You buy a new car for 24,000 dollars. 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 $n$ 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 dollars. 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 $n$ 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

Common Ratio

The common ratio is the multiplication factor used in a geometric sequence to obtain each successive term from its predecessor. In depreciation models, a common ratio less than one indicates the percentage by which the asset's value decreases annually.

Geometric Sequence

A geometric sequence is a set of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This sequence is useful for modeling situations where a quantity, such as a car's value, changes by a constant factor over regular intervals.

Exponential Decay

Exponential decay describes a process where the value of a quantity decreases at a rate proportional to its current value. In depreciation scenarios, like the one for a car, the car's value decreases by a consistent percentage each year, mirroring an exponential decay pattern.

Initial Value

The initial value is the starting amount or the first term of the sequence. In financial contexts, this represents the original price of an asset, such as the car's purchase price, which serves as the basis for calculating its depreciated value over time.

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