Problem 32 - Technical Mathematics with Calculus

Question

Inflation: The price of a certain house, now \(\$ 126,000,\) is expected to increase by \(5 \%\) each year. Write a GP whose terms are the value of the house at the end of each year, and find the value of the house after 5 years.

Step-by-Step Solution

Answer

\(T_5 = 126,000 \times 1.05^4 = \$153,153.21\)

Step 1: Identify Initial Value and Common Ratio

The annual percentage increase is 5%, which can be expressed as a decimal by dividing by 100. This gives us 0.05. Thus, the common ratio (r) of the geometric progression is 1 + 0.05 = 1.05 (since each year the price is 105% of the previous year).

Step 2: Write the Geometric Progression (GP)

The terms of the geometric progression can be expressed as: \(T_n = a \times r^{(n-1)}\), where \(a\) represents the first term and \(r\) the common ratio. For this problem: \(a = 126,000\) and \(r = 1.05\). Thus, the GP that describes the value of the house at the end of each year is \(126,000 \times 1.05^{n-1}\), where \(n\) is the number of years.

Step 3: Calculate the value of the house after 5 years

Substitute \(n = 5\) into the GP formula to find the value of the house after 5 years: \(T_5 = 126,000 \times 1.05^{5-1} = 126,000 \times 1.05^4\).

Key Concepts

Inflation in Mathematics

When studying economics or preparing for real-life financial planning, the concept of 'inflation in mathematics' becomes particularly relevant. Inflation refers to the rate at which the general price level of goods and services rises over time, leading to a decrease in the purchasing power of currency. It's a key factor to consider when predicting long-term costs or investments, such as the price of a home.

In mathematical problems, inflation is often modeled using the principles of geometric progression, where the value of an asset such as a house increases by a fixed percentage each year. To calculate future prices or value under inflation, we would apply a formula where the starting value is multiplied by a common ratio (reflecting the inflation rate) raised to the power of the number of time periods considered.

Common Ratio

The 'common ratio' in a geometric progression (GP) is one of the two defining characteristics of the sequence - the other being the first term, or initial value, of the series. A common ratio is a factor by which each term in the progression is multiplied to get the next term. In financial contexts, such as calculating inflation or interest, the common ratio corresponds to the growth rate per period plus one.

For instance, with a 5% annual inflation rate, the common ratio would be 1.05. This is because each year the value is expected to be 105% of the previous year's value, which mathematically translates to multiplying by 1.05. Understanding this concept is essential, as it allows us to model and solve complex financial problems using straightforward multiplication.

Geometric Series

A 'geometric series' is the sum of the terms in a geometric progression. Although in the provided exercise, we are not directly calculating the sum of a series, understanding how a single term is computed is vital in solving problems related to a geometric series. In the context of our exercise, each term represents the future value of the house.

By using the formula for the nth term of a GP, (\(T_n = a \times r^{(n-1)}\)), we can find out the value of the house after any number of years. The ability to determine the terms of a geometric series is beneficial, not just for predicting inflationary effects but also for many other applications, such as compounding interest calculations, population growth estimations, and more sophisticated financial models. Mastering the manipulation and interpretation of geometric series is an invaluable skill in both academic and real-world financial scenarios.