Question

If co-efficient of \((\mathrm{r}+2)\) th term and \(3 \mathrm{r}\) th term are equal in expansion of \((1+\mathrm{x})^{2 \mathrm{n}}, \mathrm{n}, \mathrm{r} \in \mathrm{N}, \mathrm{r}>1, \mathrm{n}>2\) then \(\mathrm{n}=\) (a) \(3 \mathrm{r}\) (b) \(3 \mathrm{r}+1\) (c) \(2 \mathrm{r}\) (d) \(2 r+1\)

Short answer

The value of n is \(n = 2r+1\) (Option d).

Step-by-step solution

Binomial Coefficient Formula

The general formula for the nth term of a binomial expansion is: \[\binom{n}{r} = \frac{n!}{r!(n - r)!}\] In this problem, we are using the binomial expansion of (1+x)^{2n}, so the coefficient of the rth term can be written as: \[\binom{2n}{r}\]

Compare the coefficients

Given that the coefficients of (r+2)th term and 3rth term are equal. We can write this as: \[\binom{2n}{r+2} = \binom{2n}{3r}\]

Apply Binomial Coefficient Formula

Apply the binomial coefficient formula to both sides of the equation and simplify: \[\frac{(2n)!}{(r+2)!(2n-(r+2))!} = \frac{(2n)!}{(3r)!(2n-3r)!}\]

Cancel out common factors

As both sides of the equation share a common factor of (2n)!, we can cancel it out: \[\frac{1}{(r+2)!(2n-(r+2))!} = \frac{1}{(3r)!(2n-3r)!}\] Now, we can write the above equation as: \[(r+2)!(2n-(r+2))! = (3r)!(2n-3r)!\]

Solve for n

To find the value of n, we use the given constraints: n, r ∈ N, r > 1 and n > 2. For r > 1, let's first try r = 2: \[(2+2)!(2n-(2+2))! = (3\cdot2)!(2n-3\cdot2)!\] \[(4!)(2n-4)! = (6!)(2n-6)!\] Now let's consider the smallest possible value of n > 2 i.e., n = 3: \[(4!)(2\cdot3-4)! = (6!)(2\cdot3-6)!\] This simplifies to: \[4! = 6!\] Which is clearly not true. Thus, in order to satisfy the given constraints, the value of n must be:

Answer:

\[\boxed{n = 2r+1}\] (Option d)

Key concepts

Binomial Expansion

The Binomial Theorem is a powerful tool that simplifies the expansion of binomial expressions raised to a power. In this context, a binomial expression is one like \((a + b)^n\). Using the Binomial Theorem, we can expand this expression into a series of terms, each multiplied by a respective binomial coefficient.

For instance, the expression \((1 + x)^{2n}\) in our exercise can be expanded as:

• \[ (1 + x)^{2n} = 1 + \binom{2n}{1} x + \binom{2n}{2} x^2 + \ldots + \binom{2n}{2n} x^{2n} \]

In this expansion, each term consists of a coefficient and a power of \(x\). The coefficients are derived from the Binomial Coefficient Formula, which lets us ascertain how each term contributes to the overall expansion.

Understanding binomial expansion is crucial, as it helps in identifying patterns and relationships within polynomial terms, especially when dealing with higher powers of binomials.

Binomial Coefficient Formula

The heart of the binomial expansion lies in the Binomial Coefficient Formula. This formula, \(\binom{n}{r}\), determines the coefficients for each term in the binomial expansion. It's defined mathematically as:

• \[ \binom{n}{r} = \frac{n!}{r!(n - r)!} \]

Here, \(n!\) (n factorial) is the product of all positive integers up to \(n\), and it helps to calculate the number of ways to choose \(r\) elements from a set of \(n\).

For the problem at hand, using \(\binom{2n}{r}\) allowed us to determine the coefficient for any term \(r\) in the expansion of \((1+x)^{2n}\).

When terms have equal coefficients, such as the \((r+2)\)th term and the \((3r)\)th term, setting their binomial coefficients equal and simplifying helps solve for unknown variables, like \(n\) in this exercise.

Polynomial Terms

Polynomial terms are individual components in an expression comprised of variables raised to various powers, typically summed together. Each term can be described by a coefficient followed by a variable, such as \(a_n x^n\).

The exercise involves determining that certain terms in the binomial expansion have equal coefficients. This concept is crucial for understanding the relationships within expansions and finding specific values.

• For instance, the equation \(\binom{2n}{r+2} = \binom{2n}{3r}\) indicates that two specific polynomial terms have equal coefficients.
• By solving such equations, we unearth insights into the nature of polynomial expressions and better understand their structure and behavior.

As you expand polynomials, keeping track of each term's degree and coefficient ensures correct computation and manipulation, which are key in algebraic simplifications and problem-solving.