The first three terms in the Binomial expansion of `(x+y)^n` are 1,56 and 1372 respectively . Find the value of x and y

Answer

To solve the problem, we need to find the values of \( x \) and \( y \) given that the first three terms in the binomial expansion of \( (x+y)^n \) are 1, 56, and 1372 respectively.

### Step 1: Identify the terms in the binomial expansion
The first three terms of the binomial expansion \( (x+y)^n \) are given by:
- \( T_1 = \binom{n}{0} x^n y^0 = x^n \)
- \( T_2 = \binom{n}{1} x^{n-1} y^1 = n x^{n-1} y \)
- \( T_3 = \binom{n}{2} x^{n-2} y^2 = \frac{n(n-1)}{2} x^{n-2} y^2 \)

Given:
- \( T_1 = 1 \)
- \( T_2 = 56 \)
- \( T_3 = 1372 \)

### Step 2: Set up equations based on the terms
From the information provided, we can set up the following equations:
1. \( x^n = 1 \) (from \( T_1 \))
2. \( n x^{n-1} y = 56 \) (from \( T_2 \))
3. \( \frac{n(n-1)}{2} x^{n-2} y^2 = 1372 \) (from \( T_3 \))

### Step 3: Solve for \( x \)
From the first equation \( x^n = 1 \), we can conclude that:
- \( x = 1 \) (since \( n \) is a positive integer)

### Step 4: Substitute \( x \) into the second equation
Now substituting \( x = 1 \) into the second equation:
\[ n \cdot 1^{n-1} \cdot y = 56 \implies n \cdot y = 56 \implies y = \frac{56}{n} \]

### Step 5: Substitute \( x \) into the third equation
Now substituting \( x = 1 \) into the third equation:
\[ \frac{n(n-1)}{2} \cdot 1^{n-2} \cdot y^2 = 1372 \implies \frac{n(n-1)}{2} \cdot y^2 = 1372 \]

### Step 6: Substitute \( y \) from Step 4 into the third equation
Substituting \( y = \frac{56}{n} \) into the third equation:
\[ \frac{n(n-1)}{2} \cdot \left(\frac{56}{n}\right)^2 = 1372 \]
This simplifies to:
\[ \frac{n(n-1)}{2} \cdot \frac{3136}{n^2} = 1372 \]
\[ \frac{3136(n-1)}{2n} = 1372 \]
Multiplying both sides by \( 2n \):
\[ 3136(n-1) = 2744n \]
\[ 3136n - 3136 = 2744n \]
\[ 392n = 3136 \]
\[ n = \frac{3136}{392} = 8 \]

### Step 7: Find \( y \)
Now substituting \( n = 8 \) back into the equation for \( y \):
\[ y = \frac{56}{8} = 7 \]

### Step 8: Conclusion
Thus, we have:
- \( x = 1 \)
- \( y = 7 \)

### Final Answer
The values of \( x \) and \( y \) are:
- \( x = 1 \)
- \( y = 7 \)

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