if value of 3+1/3+1/3+.......infinity is 3+root a /b  then a-b is

Let's find the value of the infinite continued fraction expression given by

\[
3 + \frac{1}{3 + \frac{1}{3 + \frac{1}{3 + \cdots}}}.
\]

To solve this, let's denote the infinite expression by \(x\):

\[
x = 3 + \frac{1}{3 + \frac{1}{3 + \frac{1}{3 + \cdots}}}.
\]

Since the expression is infinite, we observe that the expression inside the denominator after the initial \(3 + \frac{1}{\cdots}\) is the same as \(x\). So, we can rewrite the equation as:

\[
x = 3 + \frac{1}{x}.
\]

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

\[
x^2 = 3x + 1.
\]

Rearrange this equation to bring all terms to one side:

\[
x^2 - 3x - 1 = 0.
\]

This is a quadratic equation. To find \(x\), we use the quadratic formula:

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

where \(a = 1\), \(b = -3\), and \(c = -1\). Substituting these values, we get:

\[
x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1}.
\]

Simplify the expression:

\[
x = \frac{3 \pm \sqrt{9 + 4}}{2},
\]

\[
x = \frac{3 \pm \sqrt{13}}{2}.
\]

Since the expression \(3 + \frac{1}{3 + \frac{1}{3 + \cdots}}\) is positive, we take the positive root:

\[
x = \frac{3 + \sqrt{13}}{2}.
\]

We are given that \(x\) can be represented as \(3 + \frac{\sqrt{a}}{b}\). Let's compare this with our solution:

\[
\frac{3 + \sqrt{13}}{2}.
\]

To express this in the form \(3 + \frac{\sqrt{a}}{b}\), notice:

\[
3 + \frac{\sqrt{13}}{2}.
\]

Here, \(a = 13\) and \(b = 2\).

Now, calculate \(a - b\):

\[
a - b = 13 - 2 = 11.
\]

Thus, the value of \(a - b\) is \(\boxed{11}\).

Would you like to see more details or have any questions?

### Related Questions

1. How do you solve quadratic equations using the quadratic formula?
2. What are infinite continued fractions, and how are they useful in mathematics?
3. How do we determine the convergence of an infinite continued fraction?
4. Can we represent all real numbers as infinite continued fractions?
5. How can infinite series and continued fractions be related?

### Tip

When dealing with infinite expressions, look for repeating patterns to simplify and solve them effectively!

Learn how to solve the infinite continued fraction 3 + (1 / (3 + (1 / (3 + ...)))), represented as 3 + sqrt(13) / 2. Understand the application of the quadratic formula in solving the equation x^2 - 3x - 1 = 0. Suitable for advanced high school students and mathematics enthusiasts.

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