Question

The value of a car is given by the recursive rule \(a_1=25,600, a_n=0.86 a_{n-1}\), where \(n\) is the number of years since the car was new. Write an explicit rule for the value of the car after \(n\) years.

Short answer

The explicit rule for finding the value of the car after \(n\) years is given by \(a_n= 25,600*(0.86)^{n-1}\).

Step-by-step solution

Understand the Recursive Rule

The recursive rule represents two main components. The first component, \(a_1=25,600\), describes the initial value of the car. The second component, \(a_n=0.86 a_{n-1}\), dictates how the value changes with each passing year. It shows that the car's value decreases to 86% of the current value each year.

Express the rule in terms of \(n\)

The given rule states that the car's value in each subsequent year is 86% of the value from the previous year, i.e., 0.86 times the previous year's value. In terms of \(n\), it can be expressed by its correlation with \(n-1\). The explicit rule for \(a_2\) would be \(0.86a_1\), \(a_3\) would be \(0.86a_2\) or \(0.86*(0.86a_1)\), and so on.

Write the explicit rule

Notice a pattern when representing each year's car value in terms of \(a_1\). It can be seen that each year, the value gets multiplied by 0.86 another time. Hence, the explicit rule representing the value of the car after \(n\) years would be \(a_n= a_1*(0.86)^{n-1}\).

Key concepts

Explicit Rule

An explicit rule in algebra is an equation that allows you to calculate the term in a sequence without having to rely on the previous term. If you're trying to find the value of an item over time, like in our car depreciation problem, an explicit rule is incredibly useful because it saves you from having to calculate every single term in the sequence.

For the given exercise, the explicit rule is applicable for the value of the car after any number of years, n. The rule derived from the recursive pattern is expressed as: \( a_n = 25,600 \times (0.86)^{n-1} \) . Here, 25,600 represents the initial value of the car, and \( (0.86)^{n-1} \) represents the exponential decay as each year passes. Therefore, you can simply plug in the value of \( n \) to this formula and find the car's value after \( n \) years directly.

Geometric Sequences

A geometric sequence is a series 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. In our example, the value of the car forms a geometric sequence with the common ratio of 0.86. This means that every year, the car's value is multiplied by 0.86, representing a consistent percentage decrease.

The sequence looks like this: \(25,600, 25,600 \times 0.86, 25,600 \times 0.86^2, 25,600 \times 0.86^3, ...\) and so on. Understanding geometric sequences is crucial as it allows us to predict the future value of an asset or analyze its rate of depreciation over time. In financial contexts, recognizing such patterns can lead to more informed investment strategies or understanding market trends.

Exponential Decay

Exponential decay describes the process of a quantity decreasing at a rate proportionally related to its current value, such as the value of a new car dropping over time. In algebra, this is often represented by a geometric sequence where the common ratio is less than one.

For instance, in the case of our car valued initially at 25,600, the recursive rule \( a_n = 0.86a_{n-1} \) shows that the car's value decreases to 86% of the previous year's value, demonstrating exponential decay. This is common in various real-world situations, such as the depreciation of assets, population decline, or radioactive decay, where understanding the pattern of decay can be important for making predictions about the future or for scientific modeling.