Find the value of `3+1/(4+(1/(3+1/(4+1/(3+. . . oo)))))` is equal to

Options

A. `4+sqrt3`
B. `1.5+sqrt3`
C. `2+sqrt3`
D. `3+2sqrt3`

Answer

To solve the expression \(3 + \frac{1}{4 + \frac{1}{3 + \frac{1}{4 + \frac{1}{3 + \ldots}}}}\), we can denote the entire expression as \(K\). Thus, we have:

\[ K = 3 + \frac{1}{4 + \frac{1}{K}} \]

Now, let's simplify this step by step.

### Step 1: Set up the equation
We start with the equation we defined:

\[ K = 3 + \frac{1}{4 + \frac{1}{K}} \]

### Step 2: Simplify the inner fraction
We can rewrite the equation to isolate the inner fraction:

\[ K - 3 = \frac{1}{4 + \frac{1}{K}} \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:

\[ (K - 3) \left(4 + \frac{1}{K}\right) = 1 \]

Expanding this, we have:

\[ (K - 3) \cdot 4 + \frac{K - 3}{K} = 1 \]

This simplifies to:

\[ 4K - 12 + \frac{K - 3}{K} = 1 \]

### Step 4: Multiply through by \(K\) to eliminate the fraction
Multiplying through by \(K\) gives:

\[ 4K^2 - 12K + K - 3 = K \]

Rearranging this, we get:

\[ 4K^2 - 12K - K - 3 = 0 \]

This simplifies to:

\[ 4K^2 - 13K - 3 = 0 \]

### Step 5: Apply the quadratic formula
Using the quadratic formula \(K = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 4\), \(b = -13\), and \(c = -3\):

\[ K = \frac{13 \pm \sqrt{(-13)^2 - 4 \cdot 4 \cdot (-3)}}{2 \cdot 4} \]

Calculating the discriminant:

\[ K = \frac{13 \pm \sqrt{169 + 48}}{8} \]
\[ K = \frac{13 \pm \sqrt{217}}{8} \]

### Step 6: Evaluate the square root
We can approximate \(\sqrt{217}\):

\[ \sqrt{217} \approx 14.7 \]

Thus, we have:

\[ K \approx \frac{13 \pm 14.7}{8} \]

Calculating the two possible values:

1. \(K_1 = \frac{27.7}{8} \approx 3.4625\)
2. \(K_2 = \frac{-1.7}{8} \approx -0.2125\)

Since \(K\) must be positive, we take:

\[ K \approx 3.4625 \]

### Final Result
Thus, the value of the original expression is:

\[ K \approx 3.4625 \]