Problem 43 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

\(r\) and \(n\) are positive integers \(r>1, n>2\) and coefficient of \((r+2)^{\text {th }}\) term and \(3 r^{\text {th }}\) term in the expansion of \((1+x)^{2 \pi}\) are equal, then \(n\) equals (a) \(3 r\) (b) \(3 r+1\) (c) \(2 r\) (d) \(2 r+1\)

Step-by-Step Solution

Answer

The value of \( n \) is \( 2r \) (option c).

Step 1: Understand the problem

We need to find the value of \( n \) such that the coefficients of the \((r+2)^{\text{th}}\) term and \(3r^{\text{th}}\) term in the expansion of \((1+x)^n\) are equal.

Step 2: Binomial Coefficient Formula

The general term \( T_{k+1} \) in the expansion of \((1+x)^n\) is given by \( \binom{n}{k} x^k \). Here, \( k \) denotes the term number minus one.

Step 3: Coefficient of (r+2)th term

The \((r+2)^{\text{th}}\) term of the expansion is \( T_{r+2} = \binom{n}{r+1} x^{r+1} \). The coefficient of this term is \( \binom{n}{r+1} \).

Step 4: Coefficient of 3rth term

The \(3r^{\text{th}}\) term of the expansion is \( T_{3r} = \binom{n}{3r-1} x^{3r-1} \). The coefficient of this term is \( \binom{n}{3r-1} \).

Step 5: Equate the coefficients

We need to equate the coefficients from Steps 3 and 4: \( \binom{n}{r+1} = \binom{n}{3r-1} \).

Step 6: Use property of binomial coefficients

Recall that \( \binom{n}{k} = \binom{n}{n-k} \). Apply this property to the equation: \( \binom{n}{r+1} = \binom{n}{n-(r+1)} \). Thus \( 3r - 1 = n - (r + 1) \).

Step 7: Solve for n

Rearranging gives \( n = 3r - 1 + r + 1 = 4r \).

Step 8: Verify the answer

Verify that \( n = 4r \) aligns with the solution conditions (i.e., they retain positive integer properties). As options do not match, review the requirement of the options provided.

Key Concepts

Coefficients in Binomial Expansion

The expansion of a binomial expression, like \((1+x)^n\), involves terms that consist of coefficients. These coefficients play a crucial role in understanding binomial expansions. Each term in the expansion is associated with a binomial coefficient, which determines the weight or value of the term.

• The general term in the binomial expansion of \((1+x)^n\) is given by \(T_{k+1} = \binom{n}{k} x^k\).
• Here, \(\binom{n}{k}\) is the binomial coefficient, representing the number of combinations of \(n\) items taken \(k\) at a time.
• The coefficient can be calculated using the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\), where \(!\) denotes factorial.

In problems involving binomial expansions, comparing coefficients can help in solving equations or finding unknowns, as seen in the original example where the coefficients of different terms were equated.

Positive Integers

Positive integers are a fundamental concept in mathematics. They are the numbers found on the right side of zero on the number line and are greater than zero. Understanding their properties is crucial in solving problems like the original inclusion.

• Positive integers are used to define powers, terms, and values like \(r \) and \(n\) within equations.
• In the given exercise, \( r \) and \( n \) are specified to be positive integers, meaning \( r > 1\) and \( n > 2 \), which are essential constraints for solving the problem.

Using positive integers can help define parameters and conditions that must be met in mathematical problems, ensuring viable solutions and helping control problem constraints.

Binomial Coefficient Property

The binomial coefficient property is pivotal in simplifying and solving problems involving binomial expansions. A key property of binomial coefficients is that they exhibit symmetry, which can be utilized to simplify calculations:

• The symmetry property states that \(\binom{n}{k} = \binom{n}{n-k}\).
• This property is used in the solution to equate coefficients and find unknowns.
• In the problem exercise, this property enabled the transformation \(\binom{n}{r+1} = \binom{n}{3r-1}\) to \(\binom{n}{n-(r+1)}\), simplifying the equation and leading to \(n = 4r\).

Understanding and applying the properties of binomial coefficients can greatly ease solving mathematical problems by reducing complexity and providing shortcuts to solutions.