Problem 8 - Technical Mathematics with Calculus

Question

Find the sum of the infinitely many terms of each GP. $$1, \frac{1}{4}, \frac{1}{16}, \dots$$

Step-by-Step Solution

Answer

The sum of the infinite GP is \( \frac{4}{3} \)

Step 1: Identify the first term and common ratio

The first term (\( a \) of the geometric progression (GP) is 1, and each term is obtained by multiplying the previous term by the common ratio (\( r \) which can be found by dividing the second term by the first term. In this case, the second term is \( \frac{1}{4} \) and the first term is 1, so \( r = \frac{1}{4} \).

Step 2: Apply the formula for the sum of an infinite GP

The sum (\( S \) of an infinite GP can be found using the formula \( S = \frac{a}{1 - r} \) where \( a \) is the first term and \( r \) is the common ratio. This formula works only if \( |r| < 1 \). Since \( r = \frac{1}{4} \) and \( |r| < 1 \) we can apply the formula.

Step 3: Calculate the sum of the infinite GP

Plugging the values into the formula, we get \( S = \frac{1}{1 - \frac{1}{4}} = \frac{1}{\frac{3}{4}} = \frac{1}{1} \times \frac{4}{3} = \frac{4}{3} \).

Key Concepts

Geometric Progression

A geometric progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the 'common ratio'. In simpler terms, it's like a chain of numbers where you get the next link by consistently applying the same multiplication step.

For example, in the GP 1, \(\frac{1}{4}\), \(\frac{1}{16}\), ... each number is one-fourth of the one before it. This steady pattern is what defines a geometric progression. Understanding this concept is pivotal as it forms the basis for analyzing patterns in various mathematical contexts, such as finance, computer science, and physics.

Common Ratio

The common ratio in a geometric progression is the factor by which we multiply each term to get the next one. Think of it as the 'DNA' of the sequence, setting up the blueprint for its growth or decay.

To find it, you simply divide any term by the previous term (after the first). In our exercise, the common ratio is found by dividing the second term, \(\frac{1}{4}\), by the first term, which is 1, yielding \(r = \frac{1}{4}\).

Remember, the behavior of the geometric progression largely depends on the value of the common ratio. If it's between -1 and 1, the terms will get closer and closer to zero as the sequence progresses. This feature is essential when dealing with infinite series, as it determines whether a sum exists.

Infinite Series

When we talk about an infinite series, we're looking at the sum of all terms in an endless sequence. It seems tricky because how can you add up an infinite number of things? But in a geometric series with a common ratio whose absolute value is less than 1, something special happens.

The terms get smaller and smaller, approaching zero, which means they add up to a finite sum. This sum can be calculated using the formula \(S = \frac{a}{1 - r}\), where \(S\) is the sum, \(a\) is the first term, and \(r\) is the common ratio. For the GP in our exercise, since \(|r| < 1\), the sum of this infinite series converges, and we find that it equals \(\frac{4}{3}\).

This concept is not just a mathematical curiosity—it has practical applications in areas like economics, physics, and engineering, where it is used to model phenomena that exhibit continuous growth or decay.