`y=sqrt(7+sqrt(7+sqrt(7+........oo)))`, then which of the following is true?

Options

A. y = 3
B. 3 lt y lt 3.5
C. y = 7
D. `gt4`

Answer

To solve the equation \( y = \sqrt{7 + \sqrt{7 + \sqrt{7 + \ldots}}} \), we can follow these steps:

### Step 1: Set up the equation
We start by recognizing that the expression inside the square root is the same as \( y \). Therefore, we can write:
\[ y = \sqrt{7 + y} \]

### Step 2: Square both sides
To eliminate the square root, we square both sides of the equation:
\[ y^2 = 7 + y \]

### Step 3: Rearrange the equation
Next, we rearrange the equation to bring all terms to one side:
\[ y^2 - y - 7 = 0 \]

### Step 4: Solve the quadratic equation
Now, we will use the quadratic formula to solve for \( y \). The quadratic formula is given by:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our case, \( a = 1 \), \( b = -1 \), and \( c = -7 \). Plugging these values into the formula gives:
\[ y = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-7)}}{2 \cdot 1} \]
\[ y = \frac{1 \pm \sqrt{1 + 28}}{2} \]
\[ y = \frac{1 \pm \sqrt{29}}{2} \]

### Step 5: Determine the positive solution
Since \( y \) represents a length (as it is derived from a square root), we only consider the positive solution:
\[ y = \frac{1 + \sqrt{29}}{2} \]

### Step 6: Approximate the value
For practical purposes, we can approximate \( \sqrt{29} \):
\[ \sqrt{29} \approx 5.385 \]
Thus,
\[ y \approx \frac{1 + 5.385}{2} \approx \frac{6.385}{2} \approx 3.1925 \]

### Conclusion
The value of \( y \) is approximately 3.1925, which suggests that the closest integer value is 3. Therefore, the correct answer is:
\[ \text{Option A: } y = 3 \]