Problem 39 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The value of the sum of the series \(3^{n} C_{0}-8^{n} C_{1}+\) \(13^{\text {" }} C_{2}-18^{n} C_{3}+\ldots\) upto \((n+1)\) terms is (A) 0 (B) \(3^{n}\) (B) \(5^{\text {" }}\) (D) none of these

Step-by-Step Solution

Answer

The sum of the series is 0.

Step 1: Understanding the Structure of the Series

The given series is \( 3^{n} C_{0} - 8^{n} C_{1} + 13^{n} C_{2} - 18^{n} C_{3} + \ldots \) up to \((n+1)\) terms. Notice that the powers of \( n \) increase by 5 in each step and the coefficients are binomial coefficients with alternating signs.

Step 2: Conceptualizing the Alternating Series

This series involves terms that alternate between negative and positive signs. The pattern can be understood as \( \sum_{k=0}^{n} (-1)^k (5k + 3)^n C_k \), given the structure \( (3 + 5k)^n C_k \) where each term follows the mentioned arithmetic progression.

Step 3: Exploring General Binomial Expansion

Recall that a similar series \( (a-b)^n \) consists of alternating signs of binomial coefficients in its expansion: \( \sum_{k=0}^{n} (-1)^k \binom{n}{k} a^{n-k} b^k \) which provides alternating positive and negative terms.

Step 4: Relating to a Binomial Series

Transform the given series into a binomial-like series by using the identity \( \binom{n+k-1}{k} \equiv C_k \). Then look for transformation into a typical expansion.

Step 5: Conclusion through Binomial Identity

Since the series has terms from \( (3 - 8 + 13 - 18 \ldots)^n) C_k \), its simplification using the reflection of the binomial theorem over the sum suggests it reduces to zero as the positive and negative terms cancel each other out.

Key Concepts

Alternating Series

An alternating series is a series with terms that change signs from positive to negative or vice versa in a sequential pattern. This kind of series is particularly interesting because of the way in which the signs affect the convergence and sum of the series. In our example, the series is composed in such a way:

• The terms switch between positive and negative.
• The formula to express this alternation is often seen as \( \sum_{k=0}^{n} (-1)^k \cdot a_k \), where each \( a_k \) term can be any expression or coefficient.

The key to understanding an alternating series is recognizing that every other term is negated. In many cases, as in the given exercise, the terms are set in a way that opposing signs help cancel out components, potentially leading to simplification such as a sum of zero. Understanding this concept is crucial when approaching problems involving alternating series.

Binomial Coefficients

Binomial coefficients are central to combinatorics and have numerous applications, especially in the binomial theorem. These coefficients, often represented in the form \( C_k = \binom{n}{k} \), appear in the expansion of binomial expressions.

• A binomial coefficient \( \binom{n}{k} \) gives the number of ways to choose \( k \) elements from a set of \( n \) elements without regard to order.
• They can be calculated using the formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \).

In the context of the given series, these coefficients help determine each term of the series. Each term’s coefficient is a binomial coefficient that influences its magnitude. When used in an alternating series, binomial coefficients dictate the alternating pattern of the series’ terms, with each term being conditioned by \( (-1)^k \cdot C_k \). This blending of alternating signs and binomial coefficients underpins the series’ ultimate behavior and sum.

Arithmetic Progression

An arithmetic progression (AR) is a sequence of numbers in which the difference between consecutive terms is constant. The relevance of arithmetic progression becomes clear when examining the exponents or the bases in our series.

• The series given in our exercise showcases numbers whose exponential base is part of an arithmetic progression. For instance, the base values \( 3, 8, 13, 18 \ldots \) differ by a constant: 5.
• Mathematically, an AR is expressed as \( a, a + d, a + 2d, a + 3d, \ldots \), where \( a \) is the first term and \( d \) is the common difference.

In exercises and problems involving exponents undergoing arithmetic progression, this sequence can determine the trend or pattern the series follows. Furthermore, when combined with alternating signs, as seen in the original problem, the arithmetic progression supports a predictable, calculable change in terms, creating a more accessible form of evaluating each term and the series as a whole.