Problem 49 - Beginning and Intermediate Algebra

Question

A home purchased for \(\$ 160,000\) increases in value by \(4 \%\) per year. a) Find the general term of the geometric sequence that models the future value of the house. b) How much is the home worth 5 yr after it is purchased? (Hint: Think carefully about what number to substitute for \(n .\) ) Round the answer to the nearest dollar. (IMAGE CANT COPY)

Step-by-Step Solution

Answer

The general term of the geometric sequence representing the value of the house is given by: \(a_n = 160000 \times (1.04)^{n-1}\). After 5 years, the home will be worth approximately \(\$185,939\).

Step 1: Identify the initial value and common ratio of the geometric sequence

The initial value of the house, denoted as \(a_1\), is given as \(\$160,000\). The house's value increases by \(4\%\) per year, so the common ratio, denoted as \(r\), is \(1 + 0.04 = 1.04\).

Step 2: Derive the general term of the geometric sequence

The general term of a geometric sequence can be found using the formula: \(a_n = a_1 \times r^{n-1}\), where \(a_n\) is the value of the nth term, \(a_1\) is the initial term, \(r\) is the common ratio, and \(n\) is the position of the term. We already found the values for \(a_1\) and \(r\) in step 1. So, the general term for the geometric sequence representing the value of the house is given by: \(a_n = 160000 \times (1.04)^{n-1}\)

Step 3: Determine the home value 5 years after it is purchased

We will now use the general term formula to find the value of the house 5 years after it is purchased. We'll substitute \(n = 5\) into the formula: \(a_5 = 160000 \times (1.04)^{5-1}\) \(a_5 = 160000 \times (1.04)^4\) Now, we can calculate the value of \(a_5\) and round it to the nearest dollar: \(a_5 ≈ \$185,939\) So, 5 years after the house is purchased, it will be worth approximately \(\$185,939\).

Key Concepts

Initial Value

The initial value in a geometric sequence is the starting point, which is crucial because it sets the foundation for calculating future values. In our exercise, the initial value (\(a_1\)) is the purchasing cost of the home, which is \(\$160,000\). This is the price at which the investment—in this case, a house—begins to grow.

Understanding the initial value is important because every future calculation in a geometric sequence will build on it. If you're dealing with an investment, like the home in this example, the initial value represents the original amount of money you invested or the initial purchase price.

• The larger your initial value, the higher your overall returns as the sequence progresses.
• Any changes in this initial amount can significantly impact the sequence's results over time.

It's the first stepping stone toward modeling the increase in the house's value.

Common Ratio

The common ratio in a geometric sequence determines how each term relates to the previous one by a consistent factor. In our example, the common ratio is \(1.04\), reflecting the annual increase of \(4\%\) in the home's value.

To derive the common ratio:

• Start with the percentage increase, here it’s \(4\%\).
• Convert it into decimal form, \(0.04\).
• Add 1 to this decimal: \(1 + 0.04 = 1.04\).

The common ratio is crucial as it reflects continuous growth or depreciation. If it's above 1, the sequence shows growth, as seen in this house example.
This constant factor multiplies each term to get the next, making the sequence 'geometric.' A higher common ratio means quicker growth in each term, increasing exponentially over time. Therefore, understanding and identifying the common ratio accurately impacts accurate forecasting of future values.

Future Value

Calculating the future value in a geometric sequence allows us to predict how much the sequence will be worth at a specific point. In this exercise, we want to find out what the home's value will be 5 years after purchase.
To calculate the future value:

• Use the general term formula: \(a_n = a_1 \times r^{n-1}\).
• Substitute the given values (\(a_1 = 160,000\), \(r = 1.04\), \(n = 5\)).
• Solve to find \(a_5\): \(a_5 = 160000 \times (1.04)^4\).

After performing the calculation, the home's future value, 5 years later, is approximately \(\$185,939\).
This figure is rounded to the nearest dollar for precision. Knowing how to compute future values can guide financial planning and investment decisions. It helps you estimate the growth of an investment or value of an asset over time with regular percentage increases.

General Term

The general term of a geometric sequence helps us express the sequence's terms in a concise mathematical form. It generalizes a pattern that can be used to calculate the value at any nth position in the sequence.
The formula for the general term is:\[a_n = a_1 \times r^{n-1}\]Where:

• \(a_n\) is the term at position n.
• \(a_1\) is the initial term, \(\$160,000\) in this example.
• \(r\) is the common ratio, \(1.04\).
• \(n\) is the term number you are interested in.

This formula encapsulates how each term is derived from the previous term by multiplying by a stable common ratio. It's crucial for computations where the nth position isn't immediately obvious or is far along in the sequence.
In our scenario, the general term formula indicates stability, given the consistent annual increase in house value. Its use is indispensable for forecasting and understanding the patterns of growth or decay in consecutive terms of the sequence.