If `m=sqrt(5+sqrt(5+sqrt(5+…..)))` and `n=sqrt(5-sqrt(5-sqrt(5-…..)))`, then among the following the relation between m and holds is

Options

A. `m-n+1=0`
B. `m+n-1=0`
C. `m+n+1=0`
D. `m-n-1=0`

Answer

To solve the problem, we need to find the values of \( m \) and \( n \) based on the definitions provided and then determine the relationship between them.

### Step 1: Define \( m \)

We start with the expression for \( m \):
\[ m = \sqrt{5 + \sqrt{5 + \sqrt{5 + \ldots}}} \]
This implies that:
\[ m = \sqrt{5 + m} \]

### Step 2: Square both sides to eliminate the square root

Squaring both sides gives:
\[ m^2 = 5 + m \]

### Step 3: Rearrange the equation

Rearranging the equation leads to:
\[ m^2 - m - 5 = 0 \]

### Step 4: Solve the quadratic equation for \( m \)

Using the quadratic formula \( m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = -1, c = -5 \):
\[ m = \frac{1 \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-5)}}{2 \cdot 1} \]
\[ m = \frac{1 \pm \sqrt{1 + 20}}{2} \]
\[ m = \frac{1 \pm \sqrt{21}}{2} \]

Since \( m \) must be positive, we take the positive root:
\[ m = \frac{1 + \sqrt{21}}{2} \]

### Step 5: Define \( n \)

Next, we define \( n \):
\[ n = \sqrt{5 - \sqrt{5 - \sqrt{5 - \ldots}}} \]
This implies that:
\[ n = \sqrt{5 - n} \]

### Step 6: Square both sides to eliminate the square root

Squaring both sides gives:
\[ n^2 = 5 - n \]

### Step 7: Rearrange the equation

Rearranging the equation leads to:
\[ n^2 + n - 5 = 0 \]

### Step 8: Solve the quadratic equation for \( n \)

Using the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = 1, c = -5 \):
\[ n = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-5)}}{2 \cdot 1} \]
\[ n = \frac{-1 \pm \sqrt{1 + 20}}{2} \]
\[ n = \frac{-1 \pm \sqrt{21}}{2} \]

Since \( n \) must also be positive, we take the positive root:
\[ n = \frac{-1 + \sqrt{21}}{2} \]

### Step 9: Determine the relationship between \( m \) and \( n \)

Now we have:
\[ m = \frac{1 + \sqrt{21}}{2} \quad \text{and} \quad n = \frac{-1 + \sqrt{21}}{2} \]

### Step 10: Find \( m - n \)

Calculating \( m - n \):
\[ m - n = \left(\frac{1 + \sqrt{21}}{2}\right) - \left(\frac{-1 + \sqrt{21}}{2}\right) \]
\[ m - n = \frac{1 + \sqrt{21} + 1 - \sqrt{21}}{2} = \frac{2}{2} = 1 \]

### Conclusion

Thus, we can express this as:
\[ m - n - 1 = 0 \]

This means the correct relation is:
\[ m - n - 1 = 0 \]