Problem 49 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(C_{0}, C_{1}, C_{2}, \ldots, C_{n}\) are the coefficients of the expansion of \((1+x)^{\mathrm{n}}\), then the value of \(\sum_{0}^{n} \frac{C_{k}}{k+1}\) is (A) 0 (B) \(\frac{2^{n}-1}{n}\) (C) \(\frac{2^{n+1}-1}{n+1}\) (D) none of these

Step-by-Step Solution

Answer

The value is (C) \(\frac{2^{n+1}-1}{n+1}\).

Step 1: Identify the Binomial Coefficients

The coefficients \(C_0, C_1, C_2, \ldots, C_n\) in the expansion of \((1+x)^n\) are given by the binomial coefficients \(\binom{n}{k}\). Therefore, \(C_k = \binom{n}{k}\).

Step 2: Set Up the Sum

We need to find the value of the sum \(\sum_{k=0}^{n} \frac{C_k}{k+1}\). This translates to \(\sum_{k=0}^{n} \frac{1}{k+1} \binom{n}{k}\).

Step 3: Use Known Binomial Identity

We use the identity \(\sum_{k=0}^{n} \frac{1}{k+1} \binom{n}{k} = \frac{2^{n+1}-1}{n+1}\). This is a well-known result in combinatorics related to averaging binomial coefficients.

Step 4: Check Against the Given Options

From the identity, the result of the sum is \(\frac{2^{n+1}-1}{n+1}\), which matches option (C). Therefore, the correct option is (C).

Key Concepts

Combinatorics

Combinatorics is a fascinating branch of mathematics that deals with counting, arranging, and analyzing finite structures.
Understanding combinatorics helps us solve problems involving combinations and permutations. In this context, combinations refer to selecting items from a larger set, where order does not matter.

• A key concept in combinatorics is the use of binomial coefficients to count combinations.


• Binomial coefficients, written as \(\binom{n}{k}\), represent the number of ways to choose \(k\) items from \(n\) items without considering the order.

The sum \(\sum_{k=0}^{n} \frac{C_k}{k+1}\) incorporates these coefficients, making it a combinatorial problem.
The sum essentially takes an average of some form of the binomial coefficients, which is a common technique in combinatorial analysis.

Binomial Theorem

The Binomial Theorem provides a powerful tool for expanding expressions raised to a power.
This theorem states that any power of a binomial, such as \((1+x)^n\), can be expanded into a sum of terms involving binomial coefficients.

• The general formula is \((1+x)^n = \sum_{k=0}^{n} \binom{n}{k}x^k\).


• Each coefficient \(\binom{n}{k}\) in the expansion represents the number of ways to choose \(k\) items from \(n\), and the terms include powers of \(x\). This is crucial in both algebra and number theory applications.

In the context of the given exercise, the Binomial Theorem helps us understand the structure of the sum involving binomial coefficients.
The specific identity \(\sum_{k=0}^{n} \frac{1}{k+1} \binom{n}{k} = \frac{2^{n+1}-1}{n+1}\) is derived from manipulating expressions generated by the Binomial Theorem.

Algebra

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and expressions.
It is fundamental for articulating and solving equations, as well as for understanding mathematical structures.

• In algebra, we frequently encounter polynomials and their expansions, such as \((1+x)^n\), which is directly addressed by the Binomial Theorem.


• Working with algebraic identities, like \(\sum_{k=0}^{n} \frac{1}{k+1} \binom{n}{k} = \frac{2^{n+1}-1}{n+1}\), involves manipulating these symbols to simplify expressions and reach the solutions.

Understanding algebraic principles is essential for analyzing the relationships between numbers and solving complex problems like the one in the exercise.
Algebra provides the tools to handle symbolic manipulations, making it an indispensable part of solving the original problem of identifying the correct value of the sum in the given options.