Problem 50 - Beginning and Intermediate Algebra

Question

A home purchased for \(\$ 140,000\) increases in value by \(5 \%\) 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 8 yr after it is purchased? (Hint: Think carefully about what number to substitute for \(n .\) ) Round the answer to the nearest dollar.

Step-by-Step Solution

Answer

The general term of the geometric sequence that models the future value of the house is \(t_n = (\$140,000) \cdot (1.05)^n\). The value of the house after 8 years is approximately \(\$205,107.90\).

Step 1: Identifying the Initial Value, Common Ratio, and Term Number

In this case, the initial value, \(a\), is the cost of the house, which is \(\$140,000\). The common ratio, \(r\), can be represented as the percentage increase in value plus 1, or \(1 + 5\% = 1.05\). For part b, the term number, \(n\), is equal to the number of years, which is 8.

Step 2: General Term of the Geometric Sequence

The general term, \(t_n\), of a geometric sequence can be found using the formula: \[t_n = a\cdot r^n\] Where: - \(t_n\) is the future value of the house after \(n\) number of years. - \(a\) is the initial value of the house. - \(r\) is the common ratio. - \(n\) is the term number (number of years).

Step 3: Finding the General Term for this Problem

We'll now substitute the given values into the formula for the general term: \[t_n = (\$140,000) \cdot (1.05)^n\] This is the general term of the geometric sequence that models the future value of the house.

Step 4: Finding the Value of the House after 8 Years

We need to find the value of the house after 8 years, so now we will substitute \(n\) with 8 into the formula: \[t_8 = (\$140,000) \cdot (1.05)^8\]

Step 5: Calculating the Value of the House after 8 Years

Now, we'll compute the value: \[t_8 = (\$140,000) \cdot (1.05)^8 \approx \$205,107.90\] The value of the house after 8 years is approximately \(\$205,107.90\) when rounded to the nearest dollar.

Key Concepts

future value

When talking about the future value of a house or any asset, we're referring to the value that the asset is expected to have at a future point in time. In the context of our geometric sequence, the future value is represented by the general term of the sequence, denoted as \( t_n \). Here:

• \( t_n \) is the calculated value of the house in the future, after a certain number of years.
• It's calculated using the formula \( t_n = a \cdot r^n \), where \( a \) is the initial value and \( r \) is the common ratio.

For example, after 8 years, our calculation of future value is based on substituting 8 for \( n \) in the general term formula, which estimates that the house's value will grow due to the annual increase.

common ratio

The common ratio in a geometric sequence is a crucial component because it determines how the sequence progresses. In our problem, it's given by \( r = 1.05 \), which stems from the annual increase of 5% in the house's value.

• To derive \( r \), you start with the percentage increase converted into decimal form, which is 0.05, and then add 1 to it. Hence, \( r = 1 + 0.05 \).
• The common ratio illustrates the multiplying factor that the house's value undergoes each year.
• It is consistent across each term in the sequence, which is why it's referred to as 'common.'

Understanding the role of the common ratio helps predict how much the house's value will escalate annually, allowing us to calculate future values accurately.

initial value

The initial value of a geometric sequence is its starting point before any changes occur, often denoted as \( a \) in formulas. In this example, the initial value is the purchase price of the house, which is \( \$140,000 \).

• \( a \) represents the value at \( n = 0 \), before the yearly 5% increase starts impacting the value.
• It's pivotal as it serves as the base upon which future values are calculated.
• Knowing \( a \) aids in forming the general term of a geometric sequence, which maps out the forecasted path of value over time.

Think of \( a \) as the foundation your calculations build upon, which crucially impacts how accurate your future model will be.

term number

The term number, denoted as \( n \), is essentially the position within the sequence that you are interested in. It helps determine how many times the common ratio has been applied to the initial value.

• In our example, \( n \) represents the number of years after the purchase, indicating the sequence's progression for 8 years.
• Correctly identifying \( n \) ensures the computation reflects the correct point in time you are predicting the value for.
• An incorrect \( n \) would lead to wrong estimates, as it dictates how many times \( r \) gets multiplied with \( a \).

Grasping the concept of term number is fundamental in forecasting values as it directly affects where you are within the sequence timeline, making calculations accurate and meaningful.