Animated Solution for Mathematics - Quadratic Equations: The value of $3+\frac{1}{4+\frac{1}{3+\frac{1}{4+\frac{1}{3+...\infty }}}}$ is equal to

A Fractal Journey

Imagine you are standing before an infinite ladder. Each step you take reveals another step, and another, stretching into the abyss of infinity.

This is the essence of the problem we are facing today:

At first glance, it looks like a chaotic, never-ending sequence of numbers. But in the world of mathematics, infinity is not chaos; it is a pattern waiting to be tamed. Let us embark on this journey to decode the structure of this infinite ladder.

The Fractal Nature

Look closely at the expression. The beauty of this problem lies in its self-similarity.

The expression starts with $3+\frac{1}{4+\dots }$. If you look past the first two terms, the sequence begins again: $3+\frac{1}{4+\dots }$.

Because this goes on to infinity, the part of the expression starting from the second $3$ is identical to the entire expression itself. It is a mathematical fractal. We define this entire infinite structure as $x$. By doing so, we are creating a container for the infinite, allowing us to manipulate it as if it were a simple variable.

The Algebraic Trap

Now that we have named our infinite beast $x$, we can perform the magic of substitution. Since the inner part is identical to the whole, we can write:

Suddenly, the infinite ladder has collapsed into a finite, manageable algebraic equation. We have successfully trapped the infinity.

Our goal now is to solve for $x$. Let us focus on the denominator: $4+\frac{1}{x}$. By finding a common denominator, we transform this into $\frac{4x+1}{x}$.

Substituting this back, our equation becomes:

The Quadratic Battle

Dividing by a fraction is the same as multiplying by its reciprocal, so the $x$ flips to the numerator:

To isolate the fraction, we subtract $3$ from both sides: $x-3=\frac{x}{4x+1}$. Now, we cross-multiply to clear the denominator:

Expanding the left side gives us $4x^2+x-12x-3=x$. Combining the terms, we arrive at the quadratic equation:

This is the heart of the problem. We use the quadratic formula $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ with $a=4$, $b=-12$, and $c=-3$.

Calculating the discriminant, we get $144+48=192$. Thus:

The Final Verdict

We have two potential roots: $x=\frac{12\pm 8\sqrt{3}}{8}$, which simplifies to $x=1.5\pm \sqrt{3}$.

But which one is the truth? Remember, our expression is a sum of positive terms. It must be positive.

Since $\sqrt{3}\approx 1.732$, the root $1.5-\sqrt{3}$ would be negative. We must reject it.

The only valid solution is $x=1.5+\sqrt{3}$. We have conquered the infinite ladder, turning a terrifying expression into a simple, elegant number.