Problem 154 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

The sum of an infinite geometric progression is 2 and the sum of the geometric progression made from the cube of the terms of this infinite series is 24 . Then find the series.

Step-by-Step Solution

Answer

The geometric progression of the series is: \(1, 1/2, 1/4, 1/8, 1/16, \ldots\)

Step 1: Identify the given information

We are given two facts: the sum, \(S\), of an infinite geometric progression is 2, and the sum, \(S_3\), of the geometric progression derived from the cube of the terms of this infinite series is 24.

Step 2: Apply the formula for the sum of an infinite geometric series

The formula for the sum, \(S\), of an infinite geometric series with first term \(a\) and common ratio \(r\) (where \(|r| < 1\)) is \(S = \frac{a}{1 - r}\). Now, equate this with 2 to get: \(\frac{a}{1 - r} = 2\)

Step 3: Apply the formula for the sum of the cubic terms series

The sum, \(S_3\), of the progression formed from the cube of the terms can be found by the formula \(S_3 = \frac{a^3}{(1 - r)^3}\). Now, equate this with 24 to get: \(\frac{a^3}{(1 - r)^3} = 24\)

Step 4: Solve for a and r

Divide the equation from Step 3 by the cube of the equation from Step 2: \(\frac{\frac{a^3}{(1 - r)^3}}{(\frac{a}{1 - r})^3} = \frac{24}{8}\). After simplifying, this results in \(a = 1\). Plugging \(a = 1\) into the equation from Step 2 gives \(r = 1/2\)

Step 5: Construct the original series

Using the found first term and ratio, the original geometric progression would then be: \(1, 1/2, 1/4, 1/8, 1/16, \ldots\)

Key Concepts

Sum of Series

In mathematics, when working with an infinite geometric series, finding the sum is a fundamental concept. An infinite geometric series is a series of numbers in which each term after the first is found by multiplying the previous one by a fixed, constant number known as the common ratio. The sum of this series can be found using a specific formula. For an infinite geometric series with a first term designated as \(a\) and a common ratio \(r\), the sum \(S\) is given by the formula:

• \( S = \frac{a}{1 - r} \)

This formula only holds if the absolute value of the common ratio \(|r|\) is less than 1. In simple terms, this condition ensures that the series approaches a finite limit as more and more terms are added.
For example, if the sum of an infinite geometric series equals 2, then we apply the formula to find the suitable values of the first term and the common ratio. It is crucial to find the values of \(a\) and \(|r|\) that meet the condition \( \frac{a}{1 - r} = 2 \). This helps in constructing the actual series for further exploration or calculations.

Common Ratio

A common ratio in a geometric series is what distinguishes each term from its predecessor. This ratio is crucial as it dictates how the series behaves and converges, or sums up, to a particular value. The common ratio is denoted by \(r\) and is calculated by dividing any term in the series by its preceding term. It remains constant throughout the series.

• Example: In the series \(1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots\), the common ratio \(r\) is \(\frac{1}{2}\).

The key property of a common ratio in the context of an infinite geometric series is that its absolute value must be less than 1 for the series to converge. This means as you move along the series, the terms get progressively smaller, tending towards zero.
Without a proper common ratio, the series will not approach a finite sum and might instead diverge, heading off into infinity. Understanding this ratio enables us to predict the series' behavior and calculate sums efficiently.

First Term

The first term of a geometric series plays a critical role in defining the entire progression. This term, often represented as \(a\), serves as the starting point from which all subsequent terms are derived by the multiplication of the common ratio \(r\). Knowing the first term is crucial for constructing the exact series and for implementing the sum formula of the series.

• In our problem, the first term is identified as \(a = 1\).

The choice of the first term influences whether an infinite geometric series will have a finite sum — a key determinant in examining series convergence. Additionally, the first term is an essential factor in both the expression for the sum of the series and any transformations of the series, such as when each term is cubed.
Ultimately, the first term together with the common ratio \(r\) forms the foundation upon which the series builds and evolves, leading to either convergence or divergence depending on the specified conditions.