If `x=1/(1+1/(1+x))` and `y=2/(2+1/(1+y))` then which of the following can be the value of `X+Y`?

Options

A. `(-sqrt(5)-sqrt(17)+3)//4`
B. `(2sqrt(5)+sqrt(17)-3)//4`
C. `(-sqrt(5)+sqrt(17)+1)//4`
D. `(sqrt(5)+sqrt(17)-1)//4`

Answer

To solve the problem, we need to find the values of \( x \) and \( y \) given the equations:

1. \( x = \frac{1}{1 + \frac{1}{1 + x}} \)
2. \( y = \frac{2}{2 + \frac{1}{1 + y}} \)

Let's break it down step by step.

### Step 1: Solve for \( x \)

Starting with the equation for \( x \):

\[ x = \frac{1}{1 + \frac{1}{1 + x}} \]

First, simplify the denominator:

\[ \frac{1}{1 + x} = \frac{1 + x}{1 + 2 + x} = \frac{1 + x}{2 + x} \]

So we can rewrite the equation as:

\[ x = \frac{1 + x}{2 + x} \]

Now, cross-multiply to eliminate the fraction:

\[ x(2 + x) = 1 + x \]

Expanding this gives:

\[ 2x + x^2 = 1 + x \]

Rearranging the equation leads to:

\[ x^2 + x - 1 = 0 \]

### Step 2: Solve the quadratic equation for \( x \)

Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):

Here, \( a = 1, b = 1, c = -1 \):

\[ x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} \]
\[ x = \frac{-1 \pm \sqrt{1 + 4}}{2} \]
\[ x = \frac{-1 \pm \sqrt{5}}{2} \]

Taking the positive root (since \( x \) must be positive):

\[ x = \frac{-1 + \sqrt{5}}{2} \]

### Step 3: Solve for \( y \)

Now, we solve for \( y \):

\[ y = \frac{2}{2 + \frac{1}{1 + y}} \]

Again, simplify the denominator:

\[ \frac{1}{1 + y} = \frac{1 + y}{1 + 2 + y} = \frac{1 + y}{2 + y} \]

So we can rewrite the equation as:

\[ y = \frac{2(2 + y)}{2(2 + y) + 1 + y} \]

Cross-multiplying gives:

\[ y(2 + y + 1) = 2(2 + y) \]

Expanding leads to:

\[ y^2 + 3y = 4 + 2y \]

Rearranging gives:

\[ y^2 + y - 4 = 0 \]

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

Using the quadratic formula again:

Here, \( a = 1, b = 1, c = -4 \):

\[ y = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-4)}}{2 \cdot 1} \]
\[ y = \frac{-1 \pm \sqrt{1 + 16}}{2} \]
\[ y = \frac{-1 \pm \sqrt{17}}{2} \]

Taking the positive root:

\[ y = \frac{-1 + \sqrt{17}}{2} \]

### Step 5: Calculate \( x + y \)

Now we can find \( x + y \):

\[ x + y = \frac{-1 + \sqrt{5}}{2} + \frac{-1 + \sqrt{17}}{2} \]
\[ x + y = \frac{-2 + \sqrt{5} + \sqrt{17}}{2} \]

### Final Answer

Thus, the value of \( x + y \) can be expressed as:

\[ x + y = \frac{\sqrt{5} + \sqrt{17} - 2}{2} \]