Problem 70 - Contemporary Precalculus

Question

Book sales (in billions of dollars) in the United States are approximated by the sequence \(\left\\{d_{n}\right\\},\) where \(d_{n}=1.89 n+\) 42.09 and \(n=4\) corresponds to 2004. (a) What were the sales in \(2007 ?\) (b) What was the total spent on books from 2001 to \(2005 ?\)

Step-by-Step Solution

Answer

Answer: The approximate book sales in 2007 were 55.32 billion dollars, and the total spent on books from 2001 to 2005 was approximately 238.80 billion dollars.

Step 1: Find the index for 2007

Since \(n=4\) corresponds to 2004, we can figure out the index for 2007 by adding 3 to the value of n, which is \(4 + 3 = 7\). So, \(n=7\) corresponds to 2007.

Step 2: Calculate book sales in 2007 (a)

Now we can substitute the value of \(n=7\) in the given formula for \(d_n\): \(d_{n}=1.89 n + 42.09\). Thus, \(d_7 = 1.89(7) + 42.09 = 13.23 + 42.09 = 55.32\). The book sales in 2007 were approximately \(55.32\) billion dollars.

Step 3: Find the indexes for 2001 to 2005

We can find the index for 2001 by subtracting 3 from the value of n for 2004, which is \(4 - 3 = 1\). So, \(n=1\) corresponds to 2001. For 2005, add 1 to the value of n for 2004, which is \(4 + 1 = 5\). Thus, the indices for 2001 to 2005 are \(1, 2, 3, 4,\) and \(5\).

Step 4: Calculate book sales from 2001 to 2005 (b)

To find the total book sales from 2001 to 2005, we will sum the values of \(d_n\) for the indices \(1, 2, 3, 4,\) and \(5\). \(d_1 = 1.89(1) + 42.09 = 43.98\) \(d_2 = 1.89(2) + 42.09 = 45.87\) \(d_3 = 1.89(3) + 42.09 = 47.76\) \(d_4 = 1.89(4) + 42.09 = 49.65\) \(d_5 = 1.89(5) + 42.09 = 51.54\) Now, add these values together: \(43.98 + 45.87 + 47.76 + 49.65 + 51.54 = 238.80\) The total spent on books from 2001 to 2005 was approximately \(238.80\) billion dollars.

Key Concepts

Arithmetic Sequences

An arithmetic sequence is a list of numbers in which each term after the first is found by adding a constant value, called the common difference, to the previous term. In the context of the exercise, the sequence \(\{d_n\}\) represents book sales over the years. Each term of the sequence is calculated by the formula \(d_n = 1.89n + 42.09\). This means the book sales increase by a constant amount of $1.89 billion each year.

Understanding arithmetic sequences is essential when analyzing patterns over time, such as financial data or population growth.

• **Term**: Each number in the sequence.
• **First Term (\(a_1\))**: In this example, \(d_1 = 43.98\).
• **Common Difference (\(d\))**: The constant amount each term increases or decreases by, which is \(d = 1.89\) in this problem.

Recognizing these components helps simplify not only computation but also understanding of data presented as sequences.

Mathematical Modeling

Mathematical modeling involves using mathematical formulas and concepts to represent real-world problems. This method helps interpret data, make predictions, and analyze trends. In the given exercise, the sales data is modeled by the arithmetic sequence \(d_n = 1.89n + 42.09\). By substituting different values of \(n\), we can predict the sales for any given year.

Mathematical models often involve:

• **Variables**: Represent quantities, such as sales or time.
• **Expressions**: Combine variables using operations to represent relationships.
• **Equations**: Set expressions equal to numbers or other expressions, as seen with \(d_n\).

This approach not only provides clarity but also guides decision-making in business and economics.

Summation of Series

Summation of series deals with adding together the terms of a sequence over a specified range. In the exercise, the total book sales from 2001 to 2005 is calculated using summation. Each term from \(d_1\) to \(d_5\) is individually found and summed.

The steps for summation include:

• **Determine Terms**: Calculate each sequence term using the formula, here \(d_n = 1.89n + 42.09\).
• **Add Terms**: Sum all the calculated terms for the given range.
• **Total**: The result, \(238.80\) billion dollars in this instance, represents the cumulative result of all terms.

This technique is common in financial projections, budgeting, and resource allocation, where aggregating figures provide insights into overall trends.