Problem 122 - A Complete Resource Book in Mathematics for JEE Main

Question

If the sum to infinity of the series \(3+(3+d) \frac{1}{4}+(3+2 d) \frac{1}{4^{2}}+\ldots\) is \(\frac{44}{9}\), then \(d=\) (A) 1 (B) 2 (C) 4 (D) None of these

Step-by-Step Solution

Answer

The answer is (D), None of these.

Step 1: Recognize the Infinite Geometric Series

The given series is in the form of an infinite geometric series where the first term is 3 and subsequent terms increase by a common difference with each term divided by successive powers of 4. The general form for the series can be expressed as: \(S = a_1 + a_2 \cdot \frac{1}{4} + a_3 \cdot \frac{1}{4^2} + \ldots \),where the first term is 3, the second term is \((3+d)\cdot\frac{1}{4}\), the third term is \((3+2d)\cdot\frac{1}{4^2}\), etc.

Step 2: Set Up the Formula for Sum to Infinity

The formula for the sum to infinity of an infinite geometric series is:\[S = \frac{a}{1 - r}\]Where:- \(S\) is the sum to infinity,- \(a\) is the first term,- \(r\) is the common ratio.In this series, the first term \(a = 3\) and common ratio \(r = \frac{1}{4}\).

Step 3: Equate the Sum to Infinity Formula

Since the given sum to infinity is \(\frac{44}{9}\), set this equal to the sum of the geometric progression:\[\frac{3}{1 - \frac{1}{4}} = \frac{44}{9}.\]

Step 4: Solve for the Equivalent Equation

Simplify the equation provided:\[\frac{3}{\frac{3}{4}} = \frac{44}{9}\]Solving this gives:\[\frac{3 \times 4}{3} = \frac{44}{9} \Rightarrow 4 = \frac{44}{9}\].This equation is balanced, confirming that only 3 was required to match the sum.

Step 5: Evaluate if Additional Terms Affect Sum

Consider the additional terms in the series which introduce \(d\). The repeated pattern of increase is contained entirely within the division by powers of 4, so any changes must neutralize this effect, meaning \(d\) has no effect.

Step 6: Conclude the Value of \(d\)

Since the solution to the equation was met without the need to adjust \(d\) (since earlier assumptions didn't require adjusting \(d\) itself to match), we do conclude \(d = 0\), which matches none of the provided answers.

Key Concepts

Sum to Infinity

In the world of mathematics, particularly series and sequences, understanding the concept of the sum to infinity is crucial. When we talk about an infinite geometric series, it involves a sequence of numbers in which each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. The infinite part means we are considering the full, endless list of numbers generated this way.

Now, the sum to infinity specifically refers to the total sum that we approach, not exceed. For convergence (the series actually summing to a finite number), the series must satisfy certain conditions. Primarily, the magnitude of the common ratio must be less than one, which means each new term is smaller than the last. For the series to converge, this decline must persist, meaning the terms shrink towards zero.

• Convergent Series: For a geometric series to converge, the common ratio must satisfy a common ratio of less than 1 in magnitude.
• Sum Formula: The sum to infinity for a geometric series is given by \[ S = \frac{a}{1 - r} \] where \(a\) is the first term and \(r\) is the common ratio.

In the problem given, we automatically assume the series is convergent since the sum to infinity is provided. The calculated sum of \( \frac{44}{9}\) solidifies that our series not only converges but approaches a specific number seamlessly as terms are added ad infinitum.

Common Ratio

The common ratio in an infinite geometric series is the heart of the progression. It is the fixed number by which each term is multiplied to get the next. This key component determines not only the behavior of the series but crucially whether it converges to a sum.

For example, let's consider our infinite geometric series:
\(3, (3+d)\frac{1}{4}, (3+2d)\frac{1}{4^2}, \ldots \)
Here, the common ratio is clearly \( \frac{1}{4} \), as each term is derived by multiplying the preceding term by \( \frac{1}{4} \).

### Key Points about Common Ratio

• Magnitude Matters: The common ratio's absolute value must be less than 1 for the series to converge to a finite sum.
• Affects Series Growth or Decline: If the ratio is positive, terms maintain their sign; if negative, terms alternate signs.
• Impact on Series Type: A ratio greater than one leads to divergence; each term is larger than the previous.

The carefully controlled nature of our exercise ensures that the series behaves predictably, allowing us to calculate the sum to infinity using the common ratio.

Geomertic Progression

A geometric progression (often simply called a geometric series) is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant, known as the common ratio. This type of progression is easy to recognize due to this regular, multiplicative relationship between consecutive terms.

### Characteristics of Geometric Progressions

• Consistent Growth or Decline: Depending on whether the common ratio is greater or less than one, the series either consistently grows or diminishes.
• Fixed Ratio Links Terms: Each term is derived from the previous one by the common ratio, creating a neat, predictable pattern that's perfect for analysis.
• Example Usage: Commonly used in mathematical modeling of real-world applications, from finance (compound interest rates) to physics (radioactive decay).

For our original series, recognizing it as a geometric progression helps us apply the sum to infinity formula, knowing that only series with a common ratio's magnitude below 1 can offer us that finite total. Understanding this enables us to analyze and solve problems where infinite terms are theoretically added, demonstrating the elegant power of geometric progressions.