If the coefficients of `r^(th) , (r + 1)^(th)` and `(r + 2)^(th)` terms in the binomial expansion of `(1 + y)^m` are in A.P. then find the equation m and r satisfy

Options

A. `m^2 - m (4r - 1) + 4r^2 + 2 = 0`
B. `m^2 - m(4r + 1) + 4r^2 - 2 = 0`
C. `m^2 -m (4r + 1) + (4r^2 + 2) = 0`
D. `m^2 - m (4r - 1) + 4r^2 - 2 = 0`

Answer

To solve the problem, we need to find the relationship between \( m \) and \( r \) given that the coefficients of the \( r^{th} \), \( (r + 1)^{th} \), and \( (r + 2)^{th} \) terms in the binomial expansion of \( (1 + y)^m \) are in Arithmetic Progression (A.P.).

### Step-by-Step Solution:

1. **Identify the Terms**:
The \( r^{th} \), \( (r + 1)^{th} \), and \( (r + 2)^{th} \) terms in the binomial expansion of \( (1 + y)^m \) are given by:
\[ T_r = \binom{m}{r} y^r \]
\[ T_{r+1} = \binom{m}{r+1} y^{r+1} \]
\[ T_{r+2} = \binom{m}{r+2} y^{r+2} \]

2. **Set Up the A.P. Condition**:
For the coefficients to be in A.P., we have:
\[ 2 \cdot \binom{m}{r+1} = \binom{m}{r} + \binom{m}{r+2} \]

3. **Use the Binomial Coefficient Relationships**:
We can express \( \binom{m}{r+1} \) in terms of \( \binom{m}{r} \) and \( \binom{m}{r+2} \):
\[ \binom{m}{r+1} = \frac{m - r}{r + 1} \cdot \binom{m}{r} \]
\[ \binom{m}{r+2} = \frac{m - r - 1}{r + 2} \cdot \binom{m}{r+1} \]

4. **Substitute and Simplify**:
Substitute the expressions into the A.P. condition:
\[ 2 \cdot \frac{m - r}{r + 1} \cdot \binom{m}{r} = \binom{m}{r} + \frac{(m - r - 1)(m - r)}{(r + 2)(r + 1)} \cdot \binom{m}{r} \]

Dividing through by \( \binom{m}{r} \) (assuming \( \binom{m}{r} \neq 0 \)):
\[ 2 \cdot \frac{m - r}{r + 1} = 1 + \frac{(m - r - 1)(m - r)}{(r + 2)(r + 1)} \]

5. **Clear the Denominator**:
Multiply through by \( (r + 2)(r + 1) \):
\[ 2(m - r)(r + 2) = (r + 2)(r + 1) + (m - r - 1)(m - r) \]

6. **Expand and Rearrange**:
Expanding both sides:
\[ 2mr + 4m - 2r^2 - 4r = r^2 + 3r + 2 + m^2 - 2mr - m - r^2 + mr \]

Collecting like terms leads to:
\[ m^2 - 4mr + 4r + 2 = 0 \]

7. **Final Equation**:
Thus, the equation that \( m \) and \( r \) satisfy is:
\[ m^2 - 4mr + 4r + 2 = 0 \]