Problem 45 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Statement-1: \(\sum_{r=0}^{n}(r+1){ }^{n} C_{r}=(n+2) 2^{n-1}\) Statement-2: \(\sum_{r=0}^{n}(r+1)^{n} C_{r} x^{r}=(1+x)^{n}+\) \(+n x(1+x)^{n-1}\) (A) Statement- 1 is false, Statement- 2 is true (B) Statement- 1 is true, Statement- 2 is true, Statement- 2 is a correct explanation for Statement- 1 (C) Statement- 1 is true, Statement \(-2\) is true; Statement- 2 is not a correct explanation for Statement-1 (D) Statement- 1 is true, Statement- 2 is false

Step-by-Step Solution

Answer

(C) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

Step 1: Evaluate Statement 1

Consider the expression \( \sum_{r=0}^{n}(r+1) \binom{n}{r} \). We can rewrite \((r+1) \) as \( (n+1) - (n-r) \), then separate the sums. This transforms the expression\[\sum_{r=0}^{n}(r+1)\binom{n}{r} = (n+1)\sum_{r=0}^{n}\binom{n}{r} - \sum_{r=0}^{n} (n-r)\binom{n}{r} \].Using the identity \( \sum_{r=0}^{n}\binom{n}{r} = 2^n \) and calculating using the binomial theorem and differentiation, it simplifies to \((n+2)2^{n-1}\). Therefore, Statement 1 is true.

Step 2: Evaluate Statement 2

The expression \( \sum_{r=0}^{n}(r+1)\binom{n}{r}x^r \) involves binomial expansions. Expand it step-by-step:\[(1+x)^n + nx(1+x)^{n-1} = \sum_{r=0}^{n} \binom{n}{r} x^r + n \sum_{r=0}^{n-1} \binom{n-1}{r} x^{r+1} \].When simplified, it transforms into the same expression \( \sum_{r=0}^{n} (r+1) \binom{n}{r} x^r \), which aligns with the given expression. Thus, Statement 2 is true.

Step 3: Determine Explanation Relationship

Statement 2 applies when using binomial expansion and differentiation related to Statement 1. However, within the context of directly deriving \( \sum_{r=0}^{n}(r+1)\binom{n}{r} = (n+2)2^{n-1} \), it does not serve as a direct explanation. Statement 2 shows the detailed expansion but not how it explains converting analytical sums in Statement 1. Therefore, Statement 2 is not a correct explanation for Statement 1.

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics focused on counting, arrangement, and combination of sets. It is fundamental in solving problems where the arrangement of objects is essential. For instance, if you're asked to find all possible seating arrangements for a group of people, you are dealing with a combinatorial problem. When solving problems with combinatorics, we often use combinations and permutations.

• Permutations: These are arrangements where order matters. If you have a set of items and you're looking to order them, permutations come into play.
• Combinations: These are selections where order doesn't matter. For example, choosing two fruits out of five without caring about the order you pick them in.

In the context of the binomial theorem, combinatorics helps us understand the coefficients in binomial expansions. The use of combinations, specifically binomial coefficients, allows us to determine the coefficients in each term of the expansion.

Mathematical Proof

A mathematical proof is a logical argument that establishes the truth of a mathematical statement. It is composed of a series of statements and conclusions based on assumed truths and previously proven theorems. Learning the structure of mathematical proofs is essential to understanding and validating mathematical statements. There are several types of proofs:

• Direct Proof: This type involves straightforward arguments that follow logically from assumptions and known facts.
• Inductive Proof: This involves showing that if a statement holds for a base case and holds for an arbitrary step, then it holds for all numbers down the line.
• Proof by Contradiction: This technique assumes the opposite of the desired conclusion and shows that this assumption leads to a contradiction, thereby proving the original statement true.

In the exercise, we use logical steps and previously known identities like the binomial theorem to prove the correctness of given mathematical statements about sums. Breaking these down into smaller components often helps make complex relationships clear and understandable.

Binomial Coefficients

Binomial coefficients appear prominently in the binomial theorem and are crucial in expanding expressions raised to a power. They are represented as \( \binom{n}{r} \) and calculated using the combination formula:\[\binom{n}{r} = \frac{n!}{r!(n-r)!}\]where \(!\) denotes factorial, the product of all positive integers up to that number.These coefficients indicate the number of ways to choose \( r \) items from \( n \) items without considering order. When you expand \( (a + b)^n \), the coefficients of each term align with binomial coefficients. For instance, if you have \( (a + b)^2 = a^2 + 2ab + b^2 \), the coefficient 2 is \( \binom{2}{1} \), representing two ways to choose one \( b \) from two terms.Binomial coefficients also have many properties and identities useful in proofs and expanding polynomial expressions, such as:

• Pascal's Identity: \(\binom{n}{r} = \binom{n-1}{r-1} + \binom{n-1}{r}\)
• Sum of Coefficients: \(\sum_{r=0}^n \binom{n}{r} = 2^n\)

In the given exercise, understanding and manipulating these coefficients were key to verifying the correctness of the mathematical statements in question.