If `a,b,c,d` be four consecutive coefficients in the binomial expansion of `(1+x)^(n)`, then value of the expression `(((b)/(b+c))^(2)-(ac)/((a+b)(c+d)))` (where `x gt 0` and `n in N`) is

Options

A. positive
B. negative
C. zero
D. depends on `n`

Explanation

To solve the given problem, we need to find the value of the expression

\[ \left(\frac{b}{b+c}\right)^2 - \frac{ac}{(a+b)(c+d)} \]

where \(a, b, c, d\) are four consecutive coefficients in the binomial expansion of \((1+x)^n\).

### Step 1: Identify the coefficients
The coefficients in the binomial expansion are given by:
- \(a = \binom{n}{r-1}\)
- \(b = \binom{n}{r}\)
- \(c = \binom{n}{r+1}\)
- \(d = \binom{n}{r+2}\)

### Step 2: Express \(b+c\) and \(c+d\)
Using the properties of binomial coefficients, we can express:
- \(b + c = \binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{r+1}\) (using Pascal's identity)
- \(c + d = \binom{n}{r+1} + \binom{n}{r+2} = \binom{n+1}{r+2}\)

### Step 3: Substitute into the expression
Now we substitute these values into the expression:

\[ \left(\frac{b}{b+c}\right)^2 = \left(\frac{\binom{n}{r}}{\binom{n+1}{r+1}}\right)^2 \]

And for the second part:

\[ ac = \binom{n}{r-1} \cdot \binom{n}{r+1} \]

Now substituting \(a + b\) and \(c + d\):

\[ (a+b)(c+d) = \binom{n}{r-1} + \binom{n}{r} \cdot \binom{n}{r+1} + \binom{n}{r+2} = \binom{n+1}{r} \cdot \binom{n+1}{r+1} \]

### Step 4: Simplify the expression
Now we can rewrite the expression:

\[ \left(\frac{\binom{n}{r}}{\binom{n+1}{r+1}}\right)^2 - \frac{\binom{n}{r-1} \cdot \binom{n}{r+1}}{\binom{n+1}{r} \cdot \binom{n+1}{r+1}} \]

### Step 5: Find a common denominator
The common denominator for both terms is \(\binom{n+1}{r+1}^2\):

\[ \frac{\binom{n}{r}^2 - \binom{n}{r-1} \cdot \binom{n}{r+1}}{\binom{n+1}{r+1}^2} \]

### Step 6: Use the identity for binomial coefficients
Using the identity \(\binom{n}{r}^2 - \binom{n}{r-1} \cdot \binom{n}{r+1} = \binom{n-1}{r-1}\):

\[ \frac{\binom{n-1}{r-1}}{\binom{n+1}{r+1}^2} \]

### Step 7: Final simplification
We can conclude that the expression simplifies to a non-negative value, and since \(n\) and \(r\) are natural numbers, the value of the entire expression is \(0\).

### Final Answer
The value of the expression is \(0\).

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