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

Question

If \(C_{0}, C_{1}, C_{2}, \ldots, C_{\mathrm{n}}\) are the coefficients of the expansion of \((1+x)^{n}\), 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

Option C: \(\frac{2^{n+1}-1}{n+1}\) is the value of the sum.

Step 1: Identify the Expansion

In the binomial expansion of \((1+x)^n\), the coefficient \(C_k\) is given by \(\binom{n}{k}\), the binomial coefficient. Therefore, the expansion can be written as \(C_0 + C_1 x + C_2 x^2 + \cdots + C_n x^n\).

Step 2: Sum of Given Expression

We need to determine the value of \(\sum_{k=0}^{n} \frac{C_{k}}{k+1}\). This is equivalent to computing \(\sum_{k=0}^{n} \frac{\binom{n}{k}}{k+1}\).

Step 3: Use Known Result

There is a known result in combinatorics that the sum \(\sum_{k=0}^{n} \frac{\binom{n}{k}}{k+1}\) can be expressed as \(\frac{2^{n+1} - 1}{n+1}\) through identities involving binomial coefficients and integration techniques.

Step 4: Select the Correct Option

Given that \(\sum_{k=0}^{n} \frac{\binom{n}{k}}{k+1} = \frac{2^{n+1} - 1}{n+1}\), this corresponds with option (C).

Key Concepts

Binomial Coefficients

Binomial coefficients are fundamental components of the binomial theorem. They are the numbers you're looking at when staring at expressions like \(\binom{n}{k}\). But what exactly do they mean?
In simple terms, binomial coefficients tell you how many ways you can pick \(k\) objects from a set of \(n\) objects. This is central to not just binomial expansions but also much of combinatorics.
Here's the formula: \[\binom{n}{k} = \frac{n!}{k!(n-k)!} \] The exclamation mark, \(n!\), represents a factorial, which is a product of all positive integers up to \(n\).
For example, if you want to determine \(\binom{5}{2}\), that would be finding how many different ways there are to choose 2 items from 5 items. Plugging into the formula gives: \[\binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4 \times 3!}{2 \times 1 \times 3!} = \frac{20}{2} = 10\] These numbers build Pascal's Triangle, a neat way to visualize binomial coefficients. When you expand \(1 + x\) raised to some power, these coefficients accompany the terms.

Combinatorics

Combinatorics is the branch of mathematics that focuses on counting, arrangement, and combination of objects. It's the underpinning of binomial coefficients because it practically deals with questions of arrangement and selection.
When dealing with a scenario involving choices and arrangements, combinatorics provides formulas to calculate possibilities, ensuring efficient analysis without the need to list everything manually.
For example:

• How many ways can you distribute assignments to a group of students?
• How many different ways can you arrange books on a shelf?
• What is the probability of drawing certain cards from a deck?

Binomial expansions rely heavily on combinatorial logic because each term in an expansion represents different possible combinations of elements from sets.
Utilizing combinatorial identities, such as the one used in step 3 of the solution, simplifies otherwise difficult problems into manageable tasks. Understanding these principles is vital for tackling complexities in both mathematics and real-world applications.

Mathematical Induction

Mathematical induction is a powerful proving technique often used in mathematics to verify statements or formulas that apply to all natural numbers.
Think of it as a domino effect—you prove that the first domino falls, and then show that if one falls, the next one must follow. Once done, you've shown all dominos will fall.
Here’s how it typically works:

• Base Case: Verify that the statement holds for the initial value, often \(n = 1\) or \(n = 0\).
• Inductive Step: Assume the statement is true for \(n = k\) (this is your induction hypothesis). Then, you prove that if the statement holds for \(n = k\), it must also hold for \(n = k + 1\).
  
  This step is crucial as it creates a chain reaction.

In the context of our exercise, mathematical induction can be used to demonstrate some of the properties of binomial coefficients as they apply to all values of \(n\) or verify various combinatorial identities.
Mastering mathematical induction provides a reliable approach to confirming truths in mathematics, offering a foundation for rigorously proving concepts across a myriad of mathematical disciplines.