Problem 82 - Precalculus: Mathematics for Calculus

Question

Price of a House The median price of a house in Orange County increases by about \(6 \%\) per year. In 2002 the median price was \(\$ 240,000 .\) Let \(P_{n}\) be the median price \(n\) years after 2002. (a) Find a formula for the sequence \(P_{n}\) (b) Find the expected median price in 2010 .

Step-by-Step Solution

Answer

(a) \(P_{n} = 240,000 \times (1.06)^n\) (b) Expected price in 2010: \$382,524.

Step 1: Understand the problem

We need to express the median price of a house as a function of the number of years after 2002, with an annual increase of 6%. We start by understanding the nature of the problem, which is fundamentally about compounded growth.

Step 2: Establish the initial condition

In the year 2002, which corresponds to \(n = 0\), the median price \(P_0\) is given as \(\$240,000\). This will serve as the base price from which we'll calculate subsequent years' prices.

Step 3: Derive the formula for the sequence

The price increases by 6% each year, which means each subsequent year's price is 106% of the previous year's price. Mathematically, this is expressed as \(P_{n} = P_0 \times (1.06)^n\). This formula will allow us to calculate the median price for any year \(n\) after 2002.

Step 4: Specific formula for 2010

We want to find the price in 2010, which is 8 years after 2002 (since 2010 - 2002 = 8). Thus, we will substitute \(n = 8\) into the formula: \(P_8 = 240,000 \times (1.06)^8\).

Step 5: Calculate the price for 2010

Now we perform the calculation: \[P_8 = 240,000 \times (1.06)^8\approx 240,000 \times 1.59385 \approx 382,524\].Thus, the expected median price in 2010 is approximately \$382,524.

Key Concepts

Sequence Formula

Understanding sequence formulas is crucial when dealing with problems related to growth or decay over time. In this case, we are dealing with compound interest, which is a type of sequence formula. The median house price increases by a certain percentage each year, which is a classic example of exponential growth. Sequence formulas like these follow the pattern of changing by a consistent rate over regular intervals. For our specific example, we start with an initial value, in this case, the median house price in 2002 which is \(240,000. The price increases by 6% every year, which means for each year, the price is multiplied by 1.06 (100% + 6%). This gives us the formula: \( P_n = P_0 \times (1.06)^n \)where:- \( P_0 \) is the initial price (\\)240,000),- \( n \) is the number of years after 2002.The exponential nature of this sequence formula results in compound growth, which is key to understanding how median prices evolve over time.

Median Price

The concept of the median price is central in real estate and statistics. It refers to the middle value in a list of numbers. In simpler terms, it means that half of the houses in Orange County are priced below this value, and the other half above. Why use median price instead of average price? - **Stability**: The median isn’t as affected by extremely high or low prices. - **Representation**: It better reflects the market equilibrium, especially in places with significant price variability. In this exercise, the median price gives us a central reference point, indicating the general pricing trend in the housing market. Using the median price helps to paint a more accurate picture of the housing affordability and market conditions over time. It is an essential tool for both economists and homeowners when analyzing market trends.

Annual Increase

Understanding the concept of an annual increase is vital in many financial and real estate calculations. An annual increase indicates the percentage by which something, such as house prices, grows each year. Here, we have a 6% increase. This means that every year, the value increases by 6%, compounded on top of the previous year's total. This type of growth can have substantial effects over time because each year's growth builds on the growth from previous years. Let's look at its significance: - **Predictability**: Knowing the annual increase helps in predicting future prices and financial planning. - **Compounding Effect**: Even small percentage increases can lead to big changes over longer periods. For a practical application, if someone is evaluating the worth of a property investment or looking to sell, understanding the annual increase is critical. It helps in setting the right expectations and making more informed financial decisions.