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What is the result of the operation in the given image?

To solve the expression shown in the image, let’s break it down step by step. The expression presented is a nested (or continued) fraction. In these kinds of problems, it’s crucial to simplify the expression from the innermost fraction outwards.

The expression given looks like:

1 + \frac{2}{1 + \frac{2}{1 + \frac{2}{\ldots}}}

This is an example of an infinite nested fraction. To solve it, we can set the entire fraction equal to a variable, say ( x ).

Thus, we have:

x = 1 + \frac{2}{1 + \frac{2}{1 + \frac{2}{\ldots}}}

By noticing the repetitive structure, the right-side expression is essentially ( x ), which gives us:

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

Now, multiply both sides by ( x ) to eliminate the fraction:

x^2 = x + 2

Rearranging this equation results in a standard quadratic equation:

x^2 - x - 2 = 0

We can solve this quadratic equation using the quadratic formula:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Here, ( a = 1 ), ( b = -1 ), and ( c = -2 ). Plugging these into the formula gives:

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

Simplify within the square root:

x = \frac{1 \pm \sqrt{1 + 8}}{2}
x = \frac{1 \pm \sqrt{9}}{2}

Since ( \sqrt{9} = 3 ):

x = \frac{1 \pm 3}{2}

This gives two potential solutions for ( x ):

  1. $$x = \frac{4}{2} = 2$$
  2. $$x = \frac{-2}{2} = -1$$

Considering the context of the problem, ( x ) represents the value of a positive expression, so we discard ( -1 ) as a valid solution. Therefore, the solution is:

x = 2

Thus, the answer to this infinite nested fraction is C) 2.