Problem 90 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(n\) is an even positive integer and \(k=\frac{3 n}{2}\), then \(\sum_{r=1}^{k}(-3)^{r-1}{\underline{\phantom{xx}}}^{3 n} C_{2 r-1}=\) (A) \(\underline{1}\) (B) \(-1\) (C) 0 (D) none of these

Step-by-Step Solution

Answer

The answer is 0.

Step 1: Understand the Problem

To solve this problem, we need to evaluate the given sum which involves binomial coefficients and alternating powers of -3. The sum is over odd-numbered binomial coefficients of a function of an even integer, expressed as \((-3)^{r-1}\binom{3n}{2r-1}\). Here, \(n\) is even and \(k = \frac{3n}{2}\). Our task is to determine which one among the given options is the sum's value.

Step 2: Re-express the Range of Summation

Since \(k = \frac{3n}{2}\), and \(n\) is an even positive integer, it implies \(k\) is also an integer. The sum goes from \(r=1\) to \(k\) which covers all odd binomial coefficients from the lower bound of the sum to the upper limit \(k\).

Step 3: Recognize a Symmetry

Notice that the sum \(\sum_{r=1}^{k}(-3)^{r-1}\binom{3n}{2r-1}\) can be related to evaluating parts of the binomial expansion of \((1 - 3)^k\). This involves recognizing that alternating terms (e.g., \(-\binom{3n}{1}, \binom{3n}{3}, \ldots \)) could contribute canceling effects when summed.

Step 4: Apply Binomial Theorem Expansion

Apply the binomial theorem to represent expressions like \((1 + (-3))^{3n}\). Then, extract the odd terms' sum factor from this expansion evaluation, because symmetry in symmetric polynomials reveals whether these terms have canceling properties.

Step 5: Evaluate the Alternating Sum

Using part of the expansion from Step 4, particularly for odd indices, evaluate if the series indeed collapses to a recognizable pattern. Calculate \((-1)^r\binom{3n}{r}\) evaluated in congruence to part of an expansion of powers \((1-1)^{3n} = 0^{3n} = 0\).

Step 6: Conclude the Evaluation

Concluding from the combination of binomial properties and alternating series cancellation, the sum for such a problem where series may reduce itself by alternative subtraction and addition is expected to be consistently related to zero as revealed from expansion symmetry.

Key Concepts

Alternating Series

The concept of alternating series is crucial in understanding the given problem. An alternating series is a sequence of terms that alternate in sign. For example, a series might be positive, then negative, then positive again, and so on. In mathematical notation, this pattern is often represented by a factor of (-1)^n in each term where n is about the position of the term in the series.
This alternating sequence plays a significant role in the problem because the terms of the series (-3)^{r-1} change signs due to the exponent of -3. Alternating series are known for their certain properties, such as the possibility of cancellation which depends on how the terms combine and interact. This is why recognizing the alternating pattern of positive and negative terms helps make predictions about the series sum, primarily if symmetry or other properties may simplify it.
In the given problem, each odd term involves the binomial coefficient and powers of -3 that contribute to the series' alternating nature, making it key to determining the overall sum.

Binomial Coefficients

Binomial coefficients are at the heart of the problem. They arise in the expansion of binomial expressions and are denoted as \( \binom{n}{k} \), which represents the number of ways to choose k elements from a set of n elements. In the context of the binomial theorem, these coefficients appear in the expansion of expressions like \((x + y)^n\).

• The coefficient \(\binom{3n}{2r-1}\) indicates selecting certain terms required for the binomial expansion.
• These coefficients help determine which terms appear in the expansion and their respective values.

In the exercise, the binomial coefficients with odd indices (\(2r-1\)) are considered, which corresponds to specific terms in the binomial expansion. These selected terms, combined with the alternating sign, define the actual contribution of each term to the sum.
Understanding how these coefficients interact with the alternating signs is crucial for evaluating the given series. This interaction is also influenced by symmetric polynomials, leading to crucial insights regarding term cancellation and sum evaluation.

Symmetric Polynomials

Symmetric polynomials have properties that influence the evaluation of the binomial series in the problem. A symmetric polynomial is essentially a polynomial invariant under any permutation of its variables. In simple terms, the polynomial remains unchanged even if the input order of its variables is altered.
In this problem, symmetric polynomial properties come into play through the symmetry also present in binomial coefficients. They highlight that certain terms in the expansion may negate each other when summed, thanks to their symmetry.
When facing polynomial expressions, recognizing symmetric properties can simplify evaluation tremendously. In our specific exercise, observing these symmetries suggests that many terms cancel out when summed up, especially when the powers of terms are aligned symmetrically. This cancellation is an important concept to solve problems involving alternating series and binomial sum properties.
Symmetric polynomials thus help understand how a seemingly complicated pattern can result in a simple expression, such as zero in this problem, due to the canceling nature of terms in the series.