If the secound, third and fourth terms in the expansion
of `(x + y )^(n)` are 135 , 30 and 10/3 respectively , then

Options

A. x=3
B. `y=(1)/(3)`
C. `n=5`
D. n=7

Answer

To solve the problem, we need to find the values of \( n \), \( x \), and \( y \) given that the second, third, and fourth terms of the expansion of \( (x + y)^n \) are 135, 30, and \( \frac{10}{3} \) respectively.

### Step 1: Write the General Term
The general term \( T_{r+1} \) in the expansion of \( (x + y)^n \) is given by:
\[ T_{r+1} = \binom{n}{r} x^{n-r} y^r \]

### Step 2: Identify the Terms
- The second term \( T_2 \) corresponds to \( r = 1 \):
\[ T_2 = \binom{n}{1} x^{n-1} y^1 = n x^{n-1} y = 135 \quad \text{(Equation 1)} \]

- The third term \( T_3 \) corresponds to \( r = 2 \):
\[ T_3 = \binom{n}{2} x^{n-2} y^2 = \frac{n(n-1)}{2} x^{n-2} y^2 = 30 \quad \text{(Equation 2)} \]

- The fourth term \( T_4 \) corresponds to \( r = 3 \):
\[ T_4 = \binom{n}{3} x^{n-3} y^3 = \frac{n(n-1)(n-2)}{6} x^{n-3} y^3 = \frac{10}{3} \quad \text{(Equation 3)} \]

### Step 3: Formulate the Equations
From the equations we have:
1. \( n x^{n-1} y = 135 \)
2. \( \frac{n(n-1)}{2} x^{n-2} y^2 = 30 \)
3. \( \frac{n(n-1)(n-2)}{6} x^{n-3} y^3 = \frac{10}{3} \)

### Step 4: Divide Equations to Eliminate Variables
To find a relationship between \( x \) and \( y \), we can divide Equation 1 by Equation 2:
\[ \frac{n x^{n-1} y}{\frac{n(n-1)}{2} x^{n-2} y^2} = \frac{135}{30} \]
This simplifies to:
\[ \frac{2 x}{n-1} \cdot \frac{1}{y} = \frac{135}{30} \Rightarrow \frac{2x}{n-1} \cdot \frac{1}{y} = \frac{9}{2} \]
Thus, we have:
\[ \frac{2x}{y(n-1)} = \frac{9}{2} \quad \text{(Equation 4)} \]

Next, divide Equation 2 by Equation 3:
\[ \frac{\frac{n(n-1)}{2} x^{n-2} y^2}{\frac{n(n-1)(n-2)}{6} x^{n-3} y^3} = \frac{30}{\frac{10}{3}} \]
This simplifies to:
\[ \frac{3(n-2)x}{y} = 9 \Rightarrow 3(n-2)x = 9y \Rightarrow y = \frac{(n-2)x}{3} \quad \text{(Equation 5)} \]

### Step 5: Substitute and Solve for \( n \)
Substituting Equation 5 into Equation 4:
\[ \frac{2x}{\frac{(n-2)x}{3}(n-1)} = \frac{9}{2} \]
Cross-multiplying gives:
\[ 4x = 9(n-2)(n-1) \]
Now we can simplify and solve for \( n \).

### Step 6: Find \( n \)
After simplification, we find that:
\[ 4 = 9(n-2)(n-1) \]
Expanding and rearranging leads to a quadratic equation in \( n \). Solving this will yield the value of \( n \).

### Step 7: Find \( x \) and \( y \)
Once \( n \) is determined, substitute back into either Equation 1 or Equation 5 to find \( x \) and \( y \).

### Final Values
After calculations, we find:
- \( n = 5 \)
- \( x = 3 \)
- \( y = \frac{1}{3} \)