Key Concepts and Formulas

• Infinite Continued Fractions: Expressions of the form $a_0 + \frac{1}{a_1 + \frac{1}{a_2 + \dots}}$ can be solved by recognizing repeating patterns and setting the expression equal to a variable.
• Quadratic Formula: The solutions to the quadratic equation $ax^2 + bx + c = 0$ are given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Step-by-Step Solution

Step 1: Representing the Continued Fraction with a Variable

Let the value of the entire continued fraction be represented by the variable $x$: $x = 3 + {1 \over {4 + {1 \over {3 + {1 \over {4 + {1 \over {3 + ....\infty }}}}}}}}$ Why this step? This allows us to manipulate the infinite expression algebraically by assigning it a finite value.

Step 2: Identifying the Repeating Pattern and Forming an Equation

Notice that the continued fraction repeats the pattern "4 + 1/(3 + ...)" infinitely. Therefore, we can rewrite the expression for $x$ as: $x = 3 + {1 \over {4 + {1 \over {3 + {1 \over {4 + {1 \over {3 + ....\infty }}}}}}}}$ $x = 3 + {1 \over {4 + {1 \over x - 3 + 3}}}$ $x = 3 + {1 \over {4 + {1 \over x}}}$ Why this step? By recognizing the repeating pattern and substituting the variable $x$, we create an equation that relates $x$ to itself. This allows us to solve for $x$.

Step 3: Simplifying the Equation to a Quadratic Form

We now simplify the equation to obtain a standard quadratic equation of the form $ax^2 + bx + c = 0$.

• Simplify the denominator: $4 + {1 \over x} = {{4x + 1} \over x}$ Why this step? Combining the terms in the denominator makes the overall expression easier to manage.

• Substitute back and simplify the complex fraction: $x = 3 + {1 \over {{{4x + 1} \over x}}}$ $x = 3 + {x \over {4x + 1}}$ Why this step? Inverting the denominator and multiplying simplifies the fraction.

• Isolate the fractional term: $x - 3 = {x \over {4x + 1}}$ Why this step? By isolating the fraction, we prepare to eliminate the denominator.

• Eliminate the denominator by cross-multiplication: $(x - 3)(4x + 1) = x$ Why this step? This removes the fraction and allows us to expand the terms.

• Expand and rearrange into a quadratic equation: $4x^2 + x - 12x - 3 = x$ $4x^2 - 11x - 3 = x$ $4x^2 - 12x - 3 = 0$ Why this step? Expanding the product and moving all terms to one side results in a standard quadratic equation.

Step 4: Solving the Quadratic Equation

We use the quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, for the equation $4x^2 - 12x - 3 = 0$, where $a=4$, $b=-12$, and $c=-3$.

• Substitute the coefficients: $x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4(4)(-3)}}{2(4)}$ Why this step? Applying the quadratic formula directly to find the roots of the equation.

• Calculate the discriminant and simplify: $x = \frac{12 \pm \sqrt{144 + 48}}{8}$ $x = \frac{12 \pm \sqrt{192}}{8}$ Why this step? Performing the arithmetic within the square root and the denominator.

• Simplify the square root: $\sqrt{192} = \sqrt{64 \times 3} = 8\sqrt{3}$.

• Substitute the simplified root back: $x = \frac{12 \pm 8\sqrt{3}}{8}$ Why this step? Simplifying the radical makes the expression easier to handle.

• Divide numerator and denominator by their greatest common divisor (4): $x = \frac{4(3 \pm 2\sqrt{3})}{4(2)}$ $x = \frac{3 \pm 2\sqrt{3}}{2}$ Why this step? Further simplifies the expression for $x$.

Step 5: Selecting the Valid Solution

The two possible values for $x$ are: $x_1 = \frac{3 + 2\sqrt{3}}{2} \quad \text{and} \quad x_2 = \frac{3 - 2\sqrt{3}}{2}$

• Analyze the nature of the continued fraction: The original continued fraction is clearly positive since all terms are positive. Furthermore, it must be greater than 3 because the first term is 3 and the rest of the fraction is positive.

• Evaluate the roots: Since $\sqrt{3} \approx 1.732$: $x_1 = \frac{3 + 2(1.732)}{2} = \frac{3 + 3.464}{2} = \frac{6.464}{2} = 3.232$ $x_2 = \frac{3 - 2(1.732)}{2} = \frac{3 - 3.464}{2} = \frac{-0.464}{2} = -0.232$

• Choose the correct root: $x_1 \approx 3.232$ is positive and greater than 3. $x_2 \approx -0.232$ is negative, which is not possible.

Therefore, we must choose $x_1$. $x = \frac{3 + 2\sqrt{3}}{2}$

• Express in the desired format: $x = \frac{3}{2} + \frac{2\sqrt{3}}{2}$ $x = 1.5 + \sqrt{3}$

Common Mistakes & Tips

• Algebraic Manipulation: Be extremely careful with algebraic manipulations. A small error can lead to an incorrect quadratic equation and wrong answer.
• Root Selection: Always check if the roots obtained from the quadratic equation make sense in the context of the original problem. Discard extraneous roots.
• Recognizing the Pattern: Correctly identifying the repeating part of the continued fraction is crucial.

Summary

The value of the given infinite continued fraction was found by setting the expression equal to $x$, recognizing the repeating pattern, and forming the equation $x = 3 + \frac{1}{4 + \frac{1}{x}}$. This was then simplified to the quadratic equation $4x^2 - 12x - 3 = 0$. Solving this quadratic equation yielded two possible solutions, and by considering the nature of the original continued fraction, the correct solution was determined to be $x = 1.5 + \sqrt{3}$.

Final Answer The final answer is \boxed{1.5 + \sqrt{3}}, which corresponds to option (A).