Problem 126 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If the coefficients of \(\mathrm{rth},(\mathrm{r}+1)\) th and \((\mathrm{r}+2)\) th terms in the binomial expansion of \((1+y) m\) are in A. P., then \(m\) and \(\mathrm{r}\) satisfy the equation \([2005]\) (A) \(m^{2}-m(4 r-1)+4 r^{2}-2=0\) (B) \(m^{2}-m(4 r+1)+4 r^{2}+2=0\) (C) \(m^{2}-m(4 r+1)+4 r^{2}-2=0\) (D) \(m^{2}-m(4 r-1)+4 r^{2}+2=0\)

Step-by-Step Solution

Answer

The correct option is (C) for the equation that \(m\) and \(r\) satisfy.

Step 1: Understand the Binomial Coefficients

In the binomial expansion of \[(1+y)^m,\] the general term is given by \[T_{k+1} = \binom{m}{k} y^k.\] Here, \(\binom{m}{k}\) is the binomial coefficient for the \(k\)-th term.

Step 2: Identify the Coefficients

The coefficients for the \(r\), \(r+1\) and \(r+2\)-th terms in the expansion are:\[\binom{m}{r}, \binom{m}{r+1}, \binom{m}{r+2}.\] These are supposed to be in Arithmetic Progression (AP).

Step 3: Condition for Arithmetic Progression

For three quantities to be in AP, twice the middle quantity is equal to the sum of the other two quantities:\[2\binom{m}{r+1} = \binom{m}{r} + \binom{m}{r+2}.\]

Step 4: Express Binomial Coefficients Using Factorials

We know:\[\binom{m}{r} = \frac{m!}{r!(m-r)!}, \, \binom{m}{r+1} = \frac{m!}{(r+1)!(m-r-1)!}, \ \binom{m}{r+2} = \frac{m!}{(r+2)!(m-r-2)!}.\]

Step 5: Substitute and Simplify the AP Condition

Substitute the expressions for the binomial coefficients into the AP condition:\[2 \cdot \frac{m!}{(r+1)!(m-r-1)!} = \frac{m!}{r!(m-r)!} + \frac{m!}{(r+2)!(m-r-2)!}.\] Simplify by canceling common terms and rearrange.

Step 6: Solve the Simplified Equation

Upon simplifying, we get:\[2(r+1)(m-r)(m-r-1) = m(r+2)(m-r)+r(m-r-1)\]After expanding and simplifying this equation, it results in polynomial form:\[m^2 - m(4r+1) + 4r^2 - 2 = 0.\]

Step 7: Compare With Given Options

Compare the simplified equation with the given options to determine the correct choice.

Key Concepts

Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers in which the difference between consecutive terms is constant. This common difference is what gives the sequence a linear nature.
APs are often seen in sequences like 2, 5, 8, 11 where the common difference is 3. When comparing terms, as in the coefficients of a binomial expansion, being in AP means that the middle term, when doubled, equals the sum of the first and third terms.

• Properties of AP: The nth term can be expressed as: \[ a_n = a + (n-1)d \]where \( a \) is the first term, \( d \) is the common difference.
• Sum of first n terms: \[ S_n = \frac{n}{2} (2a + (n-1)d) \]

In our exercise, the binomial coefficients of the \( r, (r+1), (r+2) \) terms are in AP, leading us to create an equation where twice the \((r+1)\)-th term's coefficient equals the sum of the \( r \)-th and \((r+2)\)-th terms' coefficients.

Binomial Coefficients

The coefficients in a binomial expansion are known as binomial coefficients, which are crucial in understanding how each term contributes to the expanded form.
In a binomial theorem expansion of \[(1 + y)^m\], each term follows the format \[ T_{k+1} = \binom{m}{k} y^k \].The coefficients \( \binom{m}{k} \) indicate the number of ways to select k elements from a set of m elements, which can be calculated using factorials:

• Formula: \[ \binom{m}{k} = \frac{m!}{k!(m-k)!} \]
• Special Values: \( \binom{m}{0} = 1 \) and \( \binom{m}{m} = 1 \).

These coefficients are not just numbers from a mathematical perspective but represent specific combinations, and in our problem, they must satisfy an arithmetic progression condition, offering a bridge to formulating polynomial equations.

Polynomial Equation

A polynomial equation consists of variables, exponents, and coefficients organized under the rules of algebra. These equations can represent complex relationships between variables.
The polynomial equation derived in the context of our exercise \[ m^2 - m(4r+1) + 4r^2 - 2 = 0 \] is a crucial result from simplifying the condition laid by the arithmetic progression of the binomial coefficients.

• Structure: Polynomial equations take the form \[ a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 = 0 \]
• Types: They can be linear, quadratic, cubic, etc., depending on the highest power of the variable.

This specific quadratic equation in terms of \( m \) and \( r \) offers a fundamental insight into how these variables are interrelated by the binomial theorem and arithmetic progression conditions.