If the answer of 1 on 1+1 on x is 2, how much is the answer of 1 on 1+1 on 1+1 on x?



Let's break down the expression step by step.

The expression you provided is:

\[
\frac{1}{1 + \frac{1}{1 + \frac{1}{x}}}
\]

### Step 1: Simplify the innermost part

The innermost part is \( \frac{1}{x} \), so the term becomes:

\[
1 + \frac{1}{x}
\]

### Step 2: Substitute into the next part of the expression

Now, substitute this into the next fraction:

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

Simplify the next fraction:

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

### Step 3: Substitute the simplified expression back

Now, substitute \( \frac{x}{x + 1} \) into the main expression:

\[
\frac{1}{1 + \frac{x}{x + 1}} = \frac{1}{\frac{x + 1 + x}{x + 1}} = \frac{1}{\frac{2x + 1}{x + 1}}
\]

### Step 4: Final simplification

Now, invert the fraction to complete the expression:

\[
= \frac{x + 1}{2x + 1}
\]

Thus, the simplified result of the expression \( \frac{1}{1 + \frac{1}{1 + \frac{1}{x}}} \) is:

\[
\boxed{\frac{x + 1}{2x + 1}}
\]

---

Would you like more details or have any further questions?

Here are 5 related questions to explore further:

1. What is the value of \( x \) if \( \frac{x + 1}{2x + 1} = \frac{1}{2} \)?
2. How would the expression change if we added more fractions like \( 1 + \frac{1}{1 + \frac{1}{1 + \cdots}} \)?
3. What happens to the value of the expression as \( x \to \infty \)?
4. Can you derive a general formula for such nested fractions?
5. How do you solve similar problems involving continued fractions?

**Tip**: When simplifying complex expressions, start from the innermost part and work outward step by step for clarity.

If the answer of 1 on 1+1 on x is 2, how much is the answer of 1 on 1+1 on 1+1 on x?

Learn how to simplify the nested fraction expression 1 on 1+1 on 1+1 on x. This problem involves algebraic concepts like continued fractions and rational expressions, suitable for Grades 9-12. Step-by-step breakdown with key formulas and simplification techniques provided.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation