Problem 30 - A Complete Resource Book in Mathematics for JEE Main

Question

The sum to \(n\) terms of the series \(\frac{1}{3}+\frac{5}{9}+\frac{19}{27}+\frac{65}{81}+\ldots\) is (A) \(n-\frac{\left(3^{n}-2^{n}\right)}{2^{n}}\) (B) \(n-\frac{2\left(3^{n}-2^{n}\right)}{3^{n}}\) (C) \(2^{n}-1\) (D) \(3^{n}-1\)

Step-by-Step Solution

Answer

Option (B) is correct: \(n - \frac{2(3^{n} - 2^{n})}{3^{n}}\).

Step 1: Recognize the Pattern

Observe the terms of the series: \( \frac{1}{3}, \frac{5}{9}, \frac{19}{27}, \frac{65}{81}, \ldots \). Notice that both numerators and denominators can be expressed as sequences of powers of 2 and 3 respectively.

Step 2: Identify Numerator Pattern

The numerators are 1, 5, 19, 65. They follow the pattern: \( 1 = 2^1 - 1, \ 5 = 2^3 - 3, \ 19 = 2^5 - 13, \ 65 = 2^7 - 63 \). The sequence can be expressed as: \( 2^n - f(n) \) for some function \( f(n) \).

Step 3: Identify Denominator Pattern

Denominators are 3, 9, 27, 81 which clearly follow: \( 3^1, 3^2, 3^3, 3^4, \ldots \). The general term for denominator is \( 3^n \).

Step 4: Formulate General Term

From steps 2 and 3, deduce the general term \( T_n = \frac{2^{(2n - 1)} + 1}{3^n} \). This pattern was identified from calculations in the numerators.

Step 5: Identify the Closed Sum Formula

The sum to \( n \) terms, \( S_n \), can often be expressed using known summation formulas of the series' pattern. Recognize the combination of geometric and arithmetic progression hints.

Step 6: Verify the Given Options

Given options: \( n-\frac{(3^{n}-2^{n})}{2^{n}}, \ n-\frac{2(3^{n}-2^{n})}{3^{n}}, \ 2^{n}-1, \ 3^{n}-1 \). Substitute a few initial terms to see which option matches the sum based on calculated terms.

Key Concepts

Geometric Progression

A geometric progression is a series in which each subsequent term is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio. To identify a geometric progression, look at how the terms change as they progress.
For example, the denominators in the given series: \(3, 9, 27, 81, \ldots \) are part of a geometric sequence with a common ratio of 3.
- In the general form, a geometric sequence can be written as \(a, ar, ar^2, ar^3, \ldots\), where \(a\) is the first term and \(r\) is the common ratio.
Understanding the geometric progression helps recognize patterns that allow us to derive formulas for series summation.
When dealing with summation, the formula for the sum of \(n\) terms of a geometric progression is \(S_n = a \frac{r^n - 1}{r - 1}\), provided \(r eq 1\). Analyzing the progression aids in clearer comprehension of the sequential relationships among series terms.

Arithmetic Progression

In an arithmetic progression, the difference between consecutive terms remains constant. However, the current exercise primarily focuses on a geometric progression with hints of arithmetic patterns in the numerators.
An arithmetic sequence can be detailed as \(a, a+d, a+2d, a+3d, \ldots\), where \(a\) is the first term and \(d\) is the common difference.
- The given numerators, although not strictly arithmetic, demonstrate the importance of arithmetic reasoning when combined with other series methods.
By exploring arithmetic progression properties, one can better analyze mixed series patterns, such as those present in this exercise, where different sequences are interwoven into a complex series.
Understanding both forms allows for greater versatility in assessing various series types, essential for problems that do not neatly fit into a single category.

Sequence Patterns

Identifying sequence patterns is crucial in solving advanced series problems. In the given exercise, the numerators \(1, 5, 19, 65, \ldots\) reveal intriguing patterns.
Upon examination, each term can be expressed as different powers of 2 minus certain numbers.
- Such a pattern can be articulated as \(2^n - f(n)\), where \(f(n)\) represents a functional form to be determined through deeper analysis of sequence data.
Sequence patterns involve combing through terms to identify underlying rules, aiding in formulating expressions or equations capturing the essence of these patterns.
For complex sequences, the blend of numeral manipulation and pattern recognition tools allows for deducing the series form and potential closed-form summation expressions.

Mathematical Induction

Mathematical induction is a proof technique used to establish the truth of an infinite number of statements or series expressions. It is commonly used to confirm patterns or formulas about sequences traced in the exercise.
This method involves two main steps:
- **Base Case:** Verify that the statement is true for the initial term (usually \(n=1\)).
- **Inductive Step:** Assume the statement holds for \(n=k\) and then prove it for \(n=k+1\).
By verifying these, one shows that if the statement works for one element, it works for the next, thus proving the claim for all natural numbers indefinitely.
When dealing with the given series, employing mathematical induction can support validating that the proposed formula for \(S_n\) aligns with the series sum for all terms. Induction transforms theoretical patterns into proven mathematical assertions.