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Chapter 16: Problem 23

A machine purchased for \(\$ 75,000\) is expected to decrease in value by 20\(\%\) each year. The value of the machine after \(n\) years is \(75,000(1-0.20)^{n} .\) Use the binomial theorem to express the value of the machine after 5 years in sigma notation.

Short Answer

Expert verified
The value of the machine after 5 years is $24,576.

Step by step solution

01

Expression Analysis

The problem gives us the formula for the value of the machine after \(n\) years: \(75,000(1-0.20)^{n}\). It wants us to use the binomial theorem to express this for \(n = 5\), which means we need to expand \((1-0.20)^5\) using binomial expansion.
02

Binomial Theorem Formula

Recall the binomial theorem which states that \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). Here, \(a = 1\), \(b = -0.20\), and \(n = 5\). We will express \((1-0.20)^5\) in this manner.
03

Expand the Expression

Applying the binomial theorem, we have:\[(1 - 0.20)^5 = \sum_{k=0}^{5} \binom{5}{k} (1)^{5-k}(-0.20)^k.\] Calculate the binomial coefficients, which are values of \(\binom{5}{k}\).
04

Calculate Binomial Coefficients

Calculate each coefficient:- \(\binom{5}{0} = 1\)- \(\binom{5}{1} = 5\)- \(\binom{5}{2} = 10\)- \(\binom{5}{3} = 10\)- \(\binom{5}{4} = 5\)- \(\binom{5}{5} = 1\).
05

Substitute Coefficients and Simplify

Substitute values into the expanded expression:\[(1 - 0.20)^5 = 1 - 5(0.20) + 10(0.20)^2 - 10(0.20)^3 + 5(0.20)^4 - (0.20)^5.\]Simplify each term to express as a series in sigma notation.
06

Simplify Each Term

Calculate each term:- \((0.20)^0 = 1\) results in \(1\)- \(-5(0.20)^1 = -1\)- \(10(0.20)^2 = 0.40\)- \(-10(0.20)^3 = -0.08\)- \(5(0.20)^4 = 0.0064\)- \(-(0.20)^5 = -0.00032\).
07

Express in Sigma Notation

Express the series in sigma notation:\[(1 - 0.20)^5 = \sum_{k=0}^{5} \binom{5}{k} (1)^{5-k}(-0.20)^k = 0.32768.\]
08

Calculate Value After 5 Years

The problem statement asks for the initial machine value times the binomial expansion result:\[ \text{Value after 5 years} = 75,000 \times (1-0.20)^5 = 75,000 \times 0.32768 = 24,576. \]

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Key Concepts

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

Exponential Decay
Exponential decay describes a process where the value decreases by a consistent percentage over equal time intervals. A common real-world example is the decreasing value of a machine, like the one initially valued at $75,000. Each year, the machine loses 20% of its current value. This is represented by the formula: \[75,000(1-0.20)^n\] where \(n\) is the number of years.
  • "Exponentially" indicates that the change is multiplicative, not linear.
  • The decay factor is \(1-0.20 = 0.80\). Each year, the value is 80% of the previous year's value.
Breaking down the math, if you compute the value for a single year decay: - After 1 year: \(75,000 \times 0.80 = 60,000\). - In 5 years, continuously multiplying by 0.80 gives \(75,000 \times 0.32768\), indicating why the exponential model is fitting for such problems.
Sigma Notation
Sigma notation provides a concise way to write long sums. It's a great tool in mathematics for representing series. The expression \(\sum_{k=0}^{5} \binom{5}{k} (1)^{5-k}(-0.20)^k\) is a sigma notation. Each term in the series is summed for \(k\) from 0 to 5.
  • \(\Sigma\) stands for "sum up" and is the capital Greek letter Sigma.
  • The numbers below and above \(\Sigma\) represent starting and ending values, respectively.
  • The expression to the right of \(\Sigma\) is the general term for each step (here, based on binomial theorem).
In this context, sigma notation is used to compactly express the expansion of \((1-0.20)^5\). Instead of listing each term, sigma notation sums them automatically, saving time and space while simplifying calculations.
Binomial Coefficients
Binomial coefficients \(\binom{n}{k}\) are the heart of the binomial theorem. These coefficients determine the weights of each term in the expansion of a binomial expression. For example, in the expression \((1 - 0.20)^5\):
  • \(n=5\) is the total number of terms minus one.
  • The coefficients are calculated using the formula \(\frac{n!}{k!(n-k)!}\).
For every increment of \(k\), the binomial coefficient changes: - \(\binom{5}{0} = 1\)- \(\binom{5}{1} = 5\)- \(\binom{5}{2} = 10\), and so on.These coefficients correspond to the different combinations possible in selecting terms. They govern the rise and fall in contributions of each portion of the binomial expansion, impacting the overall series. That is why they are so vital when expanding powers of binomials.

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