Animated Solution for Mathematics - Quadratic Equations: The value of $4+\frac{1}{5+\frac{1}{4+\frac{1}{5+\frac{1}{4+......\infty }}}}$

A Journey into Continued Fractions

Imagine you are standing in a room with two parallel mirrors. You look into one, and you see an infinite corridor of reflections, each one identical to the last. This is exactly what we are dealing with in this problem.

We are looking at an infinite continued fraction:

At first glance, it looks like a monster that will never end. But in mathematics, infinity is not a wall; it is a tool.

The Recursive Loop

The secret to taming this beast is to find the pattern. Look closely at the structure: we have a $4$, then a $5$, then another $4$, then another $5$. The pattern repeats perfectly.

If we define the entire expression as $x$, then the part of the fraction starting from the second $4$ is also $x$. This is the magic of self-similarity.

We can rewrite our infinite expression as a simple recursive equation:

Suddenly, the infinite has become finite. We have trapped the infinity inside a single variable.

The Algebraic Crucible

Now that we have our equation, it is time to do some heavy lifting. We need to solve for $x$.

Let's focus on the inner denominator: $5+\frac{1}{x}$. To combine these, we find a common denominator, which gives us $\frac{5x+1}{x}$.

Now, substitute this back into our main equation:

Remember your rules of fractions: dividing by a fraction is the same as multiplying by its reciprocal. So, the expression becomes:

To make this easier to handle, let's move the $4$ to the other side: $x-4=\frac{x}{5x+1}$. Now, multiply both sides by $(5x+1)$ to clear the denominator:

The Quadratic Resolution

Expand the left side carefully: $5x^2+x-20x-4=x$. Combine the like terms: $5x^2-19x-4=x$.

Finally, bring the $x$ from the right side over to the left:

We have arrived at a standard quadratic equation. To solve for $x$, we use the quadratic formula:

Here, $a=5$, $b=-20$, and $c=-4$. Plugging these in, we get:

This simplifies to $x=\frac{20\pm \sqrt{400+80}}{10}$, which is:

The Final Selection

We are almost there. We need to simplify $\sqrt{480}$. Since $480=16\times 30$, we can pull out the $4$: $\sqrt{480}=4\sqrt{30}$.

So, $x=\frac{20\pm 4\sqrt{30}}{10}$, which simplifies to:

Now, we must make a choice. We have two possible values for $x$, but only one is physically meaningful. Since our original expression is a sum of positive terms, $x$ must be positive.

The root $2-\frac{2\sqrt{30}}{5}$ is negative, so we reject it. Our final answer is:

You have just conquered an infinite fraction by turning it into a simple quadratic. Keep this logic in your toolkit; whenever you see a repeating pattern, look for the self-similarity, and you will find the path to the solution.