Problem 65 - Contemporary Precalculus

Question

A business makes a \(\$ 10,000\) profit during its first year. If the yearly profit increases by \(\$ 7500\) in each subsequent year, what will the profit be in the tenth year and what will the total profit for the first 10 years be?

Step-by-Step Solution

Answer

Answer: The profit in the 10th year will be $77,500, and the total profit for the first 10 years will be $437,500.

Step 1: Identify the Arithmetic Formula

For an arithmetic sequence, we can use the following formulas to find any term and the sum of the first n terms: - The term formula: \(a_n = a_1+d(n-1)\), where \(a_n\) is the n-th term, \(a_1\) is the first term, \(d\) is the difference, and \(n\) is the term number. - The sum formula: \(S_n = \frac{n}{2}(a_1+a_n)\), where \(S_n\) is the sum of the first n terms. Here we have \(a_1 = \$ 10,000\) and \(d = \$ 7,500\).

Step 2: Find the 10th-Year Profit

We find the 10th term using the term formula with \(n=10\): \(a_{10} = a_1+d(10-1) =\$ 10,000 + \$ 7,500(9)\) We must calculate the value of \(a_{10}\): \(a_{10} =\$ 10,000 + \$ 67,500 =\$ 77,500\) So the profit in the 10th year will be \(\$ 77,500\).

Step 3: Calculate the Total Profit for the First 10 Years

We find the sum of the first 10 terms using the sum formula with \(n=10\) and \(a_{10} =\$ 77,500\): \(S_{10} = \frac{10}{2}(a_1+a_{10}) = 5(\$ 10,000+\$ 77,500)\) We must calculate the value of \(S_{10}\): \(S_{10} = 5(\$ 87,500) = \$ 437,500\) So the total profit for the first 10 years will be \(\$ 437,500\).

Key Concepts

Arithmetic Sequence Formula

Understanding the arithmetic sequence formula is key to grasping the progression of evenly spaced numbers. This type of sequence is common in various mathematical and real-life situations, such as saving money over time or increasing production outputs.

So what exactly is an arithmetic sequence? It's a series of numbers where each term after the first is found by adding a constant, called the common difference, to the previous term. The formula to find the n-th term of an arithmetic sequence looks like this: \(a_n = a_1 + d(n - 1)\), where:\

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• \(a_n\) is the n-th term you want to find,
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• \(a_1\) is the first term in the sequence,
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• \(d\) is the common difference between terms, and
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• \(n\) is the term number.
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\Using this formula, we can calculate any term within the sequence, provided we know the first term and the common difference.

In the context of the given exercise, we were able to determine the profit in any given year by applying this simple yet powerful formula. Let's remember that consistently understanding and using the arithmetic sequence formula gives a solid foundation for solving such problems.

Sequence Term Calculation

Calculating the specific term of an arithmetic sequence involves a straightforward application of the sequence's formula. When we say we're 'calculating a sequence term', we're essentially finding the value of an item in the sequence based on its position.

Let's break down the process using the example from the exercise: the business's profit in its 10th year. Here's what we already have:\

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• The first year’s profit, which is \(a_1 = \(10,000\),
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• The common difference, which is the annual increase of \(d = \)7,500\),
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• And the term number we are trying to find, the 10th year, so \(n = 10\).
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\With these values, we use the arithmetic sequence term formula: \(a_n = a_1 + d(n - 1)\), substituting the known values to get the profit for the 10th year. Through this calculation, we reinforce our understanding of how sequence terms build upon each other through a constant increment.

Sum of Arithmetic Series

The sum of an arithmetic series is the total of all the terms in the sequence up to a certain point. This sum can be extremely useful when looking at the overall impact or result of a sequence over time, such as the total profit over 10 years in our given problem.

To calculate the sum of the first n terms in an arithmetic series, we use the following formula: \(S_n = \frac{n}{2}(a_1 + a_n)\), where:\

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• \(S_n\) is the sum of the first n terms,
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• \(n\) is the number of terms,
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• \(a_1\) is the first term, and
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• \(a_n\) is the last term.
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\In the exercise, we were tasked to find the total profit over the first 10 years. By applying the sum formula, it was simple to determine \(S_{10}\), which yielded the substantial sum of \$437,500\. This calculation is a brilliant example of how arithmetic series play a role in aggregating values over time and showcases the cumulative effect of repeated increments.