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The problem asks you to find the decimal value of the repeated fractions \( 1 + \frac{1}{8 + \frac{1}{8 + \frac{1}{8 + \cdots}}} \), which is a continued fraction structure. The hint in the problem is that \( \frac{1}{8} = 0.125 \), and you can use this to help calculate the values of these expressions.

The four expressions in the image are:

(a) \( 1 + \frac{1}{8} \)

(b) \( 1 + \frac{1}{8 + \frac{1}{8}} \)

(c) \( 1 + \frac{1}{8 + \frac{1}{8 + \frac{1}{8}}} \)

(d) \( 1 + \frac{1}{8 + \frac{1}{8 + \frac{1}{8 + \frac{1}{8}}}} \)

Let's calculate them step by step.

### (a) \( 1 + \frac{1}{8} \)
This is simply:
\[
1 + 0.125 = 1.125
\]

### (b) \( 1 + \frac{1}{8 + \frac{1}{8}} \)
First, calculate the inner fraction:
\[
8 + \frac{1}{8} = 8 + 0.125 = 8.125
\]
Now, substitute this into the outer fraction:
\[
1 + \frac{1}{8.125} = 1 + 0.12308 \approx 1.12308
\]

### (c) \( 1 + \frac{1}{8 + \frac{1}{8 + \frac{1}{8}}} \)
First, calculate the innermost fraction:
\[
8 + \frac{1}{8} = 8.125
\]
Now calculate the next level:
\[
8 + \frac{1}{8.125} = 8 + 0.12308 = 8.12308
\]
Finally, substitute this into the outer fraction:
\[
1 + \frac{1}{8.12308} = 1 + 0.123 \approx 1.123
\]

### (d) \( 1 + \frac{1}{8 + \frac{1}{8 + \frac{1}{8 + \frac{1}{8}}}} \)
First, calculate the innermost fraction:
\[
8 + \frac{1}{8} = 8.125
\]
Next level:
\[
8 + \frac{1}{8.125} = 8.12308
\]
Next:
\[
8 + \frac{1}{8.12308} = 8.123
\]
Finally:
\[
1 + \frac{1}{8.123} = 1 + 0.123 \approx 1.123
\]

### Summary:
- (a) \( 1 + \frac{1}{8} \) ≈ **1.125**
- (b) \( 1 + \frac{1}{8 + \frac{1}{8}} \) ≈ **1.12308**
- (c) \( 1 + \frac{1}{8 + \frac{1}{8 + \frac{1}{8}}} \) ≈ **1.123**
- (d) \( 1 + \frac{1}{8 + \frac{1}{8 + \frac{1}{8 + \frac{1}{8}}}} \) ≈ **1.123**

Would you like more details or further breakdowns? Here are five related questions for practice:

1. What happens to the value as more terms are added to the continued fraction?
2. How can you write the continued fraction for \( \sqrt{2} \)?
3. Can you generalize a formula for a continued fraction involving constants other than 8?
4. How does the continued fraction expansion relate to irrational numbers?
5. What is the relationship between continued fractions and convergence speed?

**Tip:** Continued fractions are often used to approximate irrational numbers and can provide excellent accuracy with just a few terms.

Suppose I told you that the fraction 1/8 when represented as a decimal is 0.125. Use this fact and any clear relationships that you can to give the decimal value of each of the following numerical expressions.

This problem involves finding the decimal value of repeated continued fractions involving 1/8, leveraging the fact that 1/8 is 0.125. Step-by-step calculation for up to four levels of fractions, suitable for students in Grades 6-8.

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