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Chapter 8: Problem 74

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 \(n\)th term of the sequence in terms of the value of your car at the end of each year.

Short Answer

Expert verified
The car is worth \$6750 after 5 years. The \(n\)th term of the sequence, representing the value of the car at the end of each year, is 24000*(\frac{3}{4})^n.

Step by step solution

01

Calculate the value of the car after 5 years

Plug in the given value \(n = 5\) into the formula for the sequence, \(a_n = 24000 * (\frac{3}{4})^n\).\[\rightarrow a_5 = 24000 * (\frac{3}{4})^5 \rightarrow a_5 = \$6750\] Thus the car is worth \$6750 after 5 years.
02

Interpret the value of \(a_5\)

The obtained value \(a_5 = \$6750\) is the worth of the car at the end of 5 years. This means if the car was sold after 5 years of purchase, it would be sold for \$6750, assuming the car depreciates according to the given rate.
03

Describe the \(n\)th term of the sequence

The \(n\)th term of the sequence can be calculated as 24000*(\frac{3}{4})^n, where \(n\) represents the number of years after the initial purchase. This gives us the value of the car at the end of each year. Specifically, the value of the car decreases by a factor of \(\frac{3}{4}\) with each passing year.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Depreciation
Depreciation is the process by which an asset loses its value over time. For cars, this is a common occurrence because they wear down, get outdated, and have reduced performance with use. When you buy a car, it starts to depreciate as soon as you drive it off the lot.

Several factors influence depreciation, including:
  • Age: Older cars tend to have lower values.
  • Mileage: More miles usually mean more wear, reducing value.
  • Condition: Well-maintained cars depreciate less.
  • Market Demand: Certain models may hold their value better.
Understanding these aspects helps you make informed decisions about buying and selling cars.
Tracking Car Value Over Time
The value of your car changes over time due to depreciation. In mathematical terms, we can model this reduction using sequences.

For example, if a car is initially worth \(24,000, the sequence given by:\[a_n = 24000 \left(\frac{3}{4}\right)^n\] helps illustrate its value each year.

Here鈥檚 how it works:
  • At year 0, the car's value is \)24,000.
  • After 1 year, it becomes \(18,000.
  • In 5 years, it's worth \)6,750.
Using this sequence gives us a clear picture of how much our car is likely worth at any point in time.
The Role of Exponential Decay
Exponential decay describes a process where something decreases at a consistent rate over time. It's different from linear decay, where the decrease is by the same amount.

In the context of car depreciation, each year the car's value is reduced by a consistent percentage, not a fixed amount.

The formula \(a_n = 24000\left(\frac{3}{4}\right)^n\) shows exponential decay:
  • \(\frac{3}{4}\) is the decay factor. Every year, the value is 75% of the previous year.
  • This percentage drop results in faster depreciation initially, slowing over time.
So, understanding exponential decay helps us predict how quickly the car's value diminishes, aiding in financial planning.

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Most popular questions from this chapter

Follow the outline on the next page to use mathematical induction to prove that $$ \begin{aligned}(a+b)^{n}=\left(\begin{array}{c}n \\\0\end{array}\right) a^{n}+\left(\begin{array}{c}n \\\1\end{array}\right) a^{n-1} b+\left(\begin{array}{c}n \\\2\end{array}\right) a^{n-2} b^{2} \\\\+\cdots+\left(\begin{array}{c}n \\\n-1\end{array}\right) a b^{n-1}+\left(\begin{array}{c}n \\\n\end{array}\right) b^{n}\end{aligned} $$ a. Verify the formula for \(n=1\) b. Replace \(n\) with \(k\) and write the statement that is assumed true. Replace \(n\) with \(k+1\) and write the statement that must be proved. c. Multiply both sides of the statement assumed to be true by \(a+b .\) Add exponents on the left. On the right, distribute \(a\) and \(b,\) respectively. d. Collect like terms on the right. At this point, you should have $$ \begin{aligned}&(a+b)^{k+1}=\left(\begin{array}{l}k \\\0\end{array}\right) a^{k+1}+\left[\left(\begin{array}{l}k \\\0\end{array}\right)+\left(\begin{array}{l}k \\\1\end{array}\right)\right] a^{k} b\\\&\begin{array}{l}+\left[\left(\begin{array}{c}k \\\1\end{array}\right)+\left(\begin{array}{c}k \\\2\end{array}\right)\right] a^{k-1} b^{2}+\left[\left(\begin{array}{c}k \\\2\end{array}\right)+\left(\begin{array}{c}k \\\3\end{array}\right)\right] a^{k-2} b^{3} \\\\+\cdots+\left[\left(\begin{array}{c}k \\\k-1\end{array}\right)+\left(\begin{array}{c}k \\\k\end{array}\right)\right] a b^{k}+\left(\begin{array}{c}k \\\k\end{array}\right) b^{k+1} \end{array}\end{aligned} $$ e. Use the result of Exercise 74 to add the binomial sums in brackets. For example, because \(\left(\begin{array}{l}n \\\ r\end{array}\right)+\left(\begin{array}{c}n \\\ r+1\end{array}\right)$$=\left(\begin{array}{l}n+1 \\ r+1\end{array}\right),\) then \(\left(\begin{array}{l}k \\ 0\end{array}\right)+\left(\begin{array}{l}k \\\ 1\end{array}\right)=\left(\begin{array}{c}k+1 \\\1\end{array}\right)\) and\(\left(\begin{array}{l}k \\ 1\end{array}\right)+\left(\begin{array}{l}k \\\2\end{array}\right)=\left(\begin{array}{c}k+1 \\ 2\end{array}\right)\) f. Because \(\left(\begin{array}{l}k \\\ 0\end{array}\right)=\left(\begin{array}{c}k+1 \\ 0\end{array}\right) \quad\) (why?) and \(\left(\begin{array}{l}k \\ k\end{array}\right)=\) \(\left(\begin{array}{l}k+1 \\ k+1\end{array}\right)\) (why?), substitute these results and the results from part (e) into the equation in part (d). This should give the statement that we were required to prove in the second step of the mathematical induction process.

Which one of the following is true? a. \(\frac{n !}{(n-1) !}=\frac{1}{n-1}\) b. The Fibonacei sequence \(1,1,2,3,5,8,13,21,34,55,89\) \(144, \ldots\) can be defined recursively using \(a_{0}=1, a_{1}=1\) \(a_{n}=a_{n-2}+a_{n-1},\) where \(n \geq 2\). c. \(\sum_{i=1}^{2}(-1)^{i} 2^{i}=0\) d. \(\sum_{i=1}^{2} a_{i} b_{i}=\sum_{i=1}^{2} a_{i} \sum_{i=1}^{2} b_{i}\)

Explain how to find a particular term in a binomial expansion without having to write out the entire expansion.

Find the term indicated in each expansion. \((x+2 y)^{10} ;\) the term containing \(y^{6}\)

What is true about the sum of the exponents on \(a\) and \(b\) in any term in the expansion of \((a+b)^{n} ?\)
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