Key Concepts and Formulas

• Continued Fractions: An expression of the form $a_0 + \frac{1}{a_1 + \frac{1}{a_2 + \frac{1}{a_3 + ...}}}$, where $a_i$ are usually integers.
• Self-Similarity: Infinite continued fractions often exhibit self-similarity, meaning a part of the fraction repeats the same structure as the whole fraction.
• Quadratic Formula: For a quadratic equation $ax^2 + bx + c = 0$, the solutions are given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Step-by-Step Solution

1. Identify the Repeating Pattern and Assign a Variable

We observe the given infinite continued fraction and recognize its repeating pattern. We aim to represent the entire expression with a variable to simplify the problem.

Let $y = 4 + {1 \over {5 + {1 \over {4 + {1 \over {5 + {1 \over {4 + ......\infty }}}}}}}}$

Notice that the expression within the denominator after '5 + ' is similar to the original expression, allowing us to rewrite the equation.

We can rewrite the equation as: $y = 4 + \frac{1}{5 + \frac{1}{y}}$

Explanation: We assign the variable y to the entire infinite continued fraction. Due to the infinite nature of the fraction, the repeating block 4 + 1/(5 + 1/(...)) appears within the fraction itself, allowing us to substitute the entire expression with y.

2. Simplify the Expression

Now, we simplify the right-hand side of the equation to eliminate the nested fraction. $y = 4 + \frac{1}{5 + \frac{1}{y}}$

First, simplify the denominator: $5 + \frac{1}{y} = \frac{5y + 1}{y}$

Substitute this back into the main equation: $y = 4 + \frac{1}{\frac{5y + 1}{y}}$

When dividing by a fraction, we multiply by its reciprocal: $y = 4 + \frac{y}{5y + 1}$

Explanation: This step involves basic algebraic manipulation to simplify the equation and prepare it for conversion into a quadratic form.

3. Formulate a Quadratic Equation

We isolate y terms and arrange the equation into the standard quadratic form $ax^2 + bx + c = 0$. $y - 4 = \frac{y}{5y + 1}$

Multiply both sides by $(5y + 1)$ to eliminate the denominator: $(y - 4)(5y + 1) = y$

Expand the left side: $5y^2 + y - 20y - 4 = y$ $5y^2 - 19y - 4 = y$

Move the y from the right side to the left side: $5y^2 - 19y - y - 4 = 0$ $5y^2 - 20y - 4 = 0$

Explanation: We have now successfully transformed the original equation into a standard quadratic equation, which we can solve using the quadratic formula.

4. Solve the Quadratic Equation using the Quadratic Formula

For a quadratic equation in the form $ax^2 + bx + c = 0$, the solutions for x are given by the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

In our equation, $5y^2 - 20y - 4 = 0$, we have: $a = 5$ $b = -20$ $c = -4$

Substitute these values into the quadratic formula: $y = \frac{-(-20) \pm \sqrt{(-20)^2 - 4(5)(-4)}}{2(5)}$ $y = \frac{20 \pm \sqrt{400 + 80}}{10}$ $y = \frac{20 \pm \sqrt{480}}{10}$

Explanation: Applying the quadratic formula allows us to find the roots of the quadratic equation, which represent potential solutions for the value of the continued fraction.

5. Simplify the Radical and Select the Valid Solution

Simplify the square root of 480: $\sqrt{480} = \sqrt{16 \times 30} = \sqrt{16} \times \sqrt{30} = 4\sqrt{30}$

Substitute this back into the expression for y: $y = \frac{20 \pm 4\sqrt{30}}{10}$

Now, consider the two possible solutions: $y_1 = \frac{20 + 4\sqrt{30}}{10}$ $y_2 = \frac{20 - 4\sqrt{30}}{10}$

Since the original continued fraction consists only of positive numbers, its value must be positive. We discard the negative root.

The valid solution is: $y = \frac{20 + 4\sqrt{30}}{10}$

Simplify the expression by dividing both terms in the numerator by 10: $y = \frac{20}{10} + \frac{4\sqrt{30}}{10}$ $y = 2 + \frac{2\sqrt{30}}{5}$

Explanation: We select the positive root because the given continued fraction consists of positive terms, and its overall value must be positive. Simplifying the expression gives us the final answer.

6. Final Answer

The value of the given infinite continued fraction is $2 + \frac{2\sqrt{30}}{5}$.

This matches option (A).

Common Mistakes & Tips

• Incorrectly Identifying Repeating Pattern: Ensure the repeating part is correctly identified. A slight error here will propagate through the entire solution.
• Discarding the Negative Root: Remember to discard the negative root when dealing with continued fractions with positive terms, as the overall value must be positive.
• Algebraic Errors: Double-check all algebraic manipulations to avoid mistakes in expanding and simplifying the equation.

Summary

To solve infinite continued fractions, we use the self-similar property. We assign a variable to the entire expression and substitute it into the repeating part. This leads to an algebraic equation, often quadratic, which we can solve. Finally, we select the valid solution based on the problem's constraints, such as positivity. The value of the given continued fraction is $2 + \frac{2\sqrt{30}}{5}$.

Final Answer The final answer is \boxed{2 + \frac{2}{5}\sqrt {30}}, which corresponds to option (A).