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

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) Both statements are true, but Statement-2 does not explain Statement-1.

Step 1: Analyze Statement-1

Statement-1 is: \( \sum_{r=0}^{n}(r+1)\binom{n}{r}=(n+2) 2^{n-1} \).We can break this down using:\( \sum_{r=0}^{n}(r+1)\binom{n}{r} = \sum_{r=0}^{n}r\binom{n}{r} + \sum_{r=0}^{n}\binom{n}{r} \).Through the identity \( \sum_{r=0}^{n}\binom{n}{r} = 2^n \) and another identity \( \sum_{r=0}^{n}r\binom{n}{r} = n 2^{n-1} \),we can infer the statement is true:\( n 2^{n-1} + 2^n = (n+2) 2^{n-1} \).

Step 2: Analyze Statement-2

Statement-2 is: \( \sum_{r=0}^{n}(r+1)\binom{n}{r} x^{r}=(1+x)^{n}+n x(1+x)^{n-1} \)This is obtained by differentiating \( (1+x)^n \) with respect to \( x \), and using \( f'(x) + f(x) x = (1+x)^n + n x (1+x)^{n-1} \).Expanding provides:\( \sum_{r=0}^{n}(r+1)\binom{n}{r}x^r \), showing this identity is true.

Step 3: Relate the Statements

To determine if Statement-2 explains Statement-1, notice that Statement-1 builds on simplifying terms within a binomial expansion, whereas Statement-2 arises from differentiating and rearranging binomial coefficients. Even though both are true, while linked by binomial principles, Statement-2 doesn't directly provide an explanation for Statement-1's form.

Step 4: Conclusion

Statement-1 is true using identities and summations of binomial coefficients. Statement-2 is also accurate through derivative manipulation and expansion identities. Statement-2 is not an explanation for Statement-1, leading us to choose an option that reflects both statements being true but independent.

Key Concepts

Binomial Coefficients

Binomial coefficients are a crucial element in combinatorics and algebra. They are used to expand expressions raised to powers, a process known as binomial expansion. Specifically, the binomial coefficient \( \binom{n}{r} \) represents the number of ways to choose \( r \) items from a total of \( n \) without considering the order of selection. These coefficients are prominently featured in Pascal's Triangle and can be calculated using the formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
In problems like the one given in the exercise, binomial coefficients help us to decode complex summations. For example, identities such as \( \sum_{r=0}^{n} \binom{n}{r} = 2^n \) are derived from the expansion of \((1 + 1)^n\). This identity illustrates how all the subsets of \( n \) elements, including the empty set, contribute to the total number of subsets, represented by \( 2^n \).
Understanding binomial coefficients is foundational because they reveal the structure of binomial expansions and enable various algebraic manipulations.

Summation Identities

Summation identities are essential tools for simplifying expressions and verifying algebraic statements. They frequently involve binomial coefficients and are often used to evaluate complex series. For instance, in our problem, the identity \( \sum_{r=0}^{n} r \binom{n}{r} = n 2^{n-1} \) combines the concept of weighted sums with binomial coefficients.
Such identities are derived using the properties of functions and algebraic manipulations such as differentiation or transformation of terms. In the context of the exercise, we used it to verify Statement-1 by expressing the sum \( \sum_{r=0}^{n} (r+1) \binom{n}{r} \) as the sum of two simpler identites. This simplification helps show that \( n 2^{n-1} + 2^n = (n+2) 2^{n-1} \), confirming the truth of the statement through rearrangement of outlined identities.
Summation identities unlock the potential of manipulating binomial expressions and help venture beyond plain expansions into areas involving calculus and number theory.

Derivative of Binomial Expression

When encountering polynomial expressions like \((1+x)^n\), taking derivatives is a powerful method used to gain insight into their behavior. Differentiation transforms a polynomial by focusing on its coefficients, revealing relationships and simplifications not immediately apparent.
For the expression \( (1+x)^n \), the derivative with respect to \( x \) is \( n(1+x)^{n-1} \). Applying this result, the exercise incorporates this derivative in Statement-2 to demonstrate the relationship established by the identity:
\[ \sum_{r=0}^{n} (r+1) \binom{n}{r} x^r = (1+x)^n + nx(1+x)^{n-1} \]
By differentiating the expanded form of the binomial expression and rearranging it with multiplication of certain terms, we derive an insightful new identity. Here, differentiation reveals hidden connections between terms that might initially appear unrelated. Through derivatives, we unlock deeper properties of expressions that enrich our understanding of algebraic structures.