What is the length of the sequence of different digits in the decimal equivalent of `(3)/(7)`?

Answer

To find the length of the sequence of different digits in the decimal equivalent of \( \frac{3}{7} \), we will follow these steps:

### Step 1: Perform the Division
We start by dividing 3 by 7 to find its decimal equivalent.

\[ 3 \div 7 = 0.428571428571... \]

### Step 2: Identify the Decimal Expansion
From the division, we can see that the decimal expansion of \( \frac{3}{7} \) is:

\[ 0.428571428571... \]

### Step 3: Determine the Repeating Sequence
Next, we observe the decimal expansion to identify the repeating sequence. The digits after the decimal point are:

\[ 428571 \]

### Step 4: Count the Unique Digits
Now, we count the number of unique digits in the repeating sequence \( 428571 \). The digits are:

- 4
- 2
- 8
- 5
- 7
- 1

### Step 5: Calculate the Length of the Sequence
The length of the sequence of different digits is the number of unique digits we found. In this case, there are 6 unique digits.

### Conclusion
Thus, the length of the sequence of different digits in the decimal equivalent of \( \frac{3}{7} \) is:

\[ \boxed{6} \]