6 349 565 ? 754 781

Options

A. 629
B. 590
C. 601
D. 690

Explanation

To solve the series 6, 349, 565, ?, 754, 781, we need to identify the pattern in the differences between the numbers. Let's break it down step by step:

### Step 1: Identify the Numbers in the Series
The series we have is:
- 6
- 349
- 565
- ?
- 754
- 781

### Step 2: Calculate the Differences Between Consecutive Terms
We will calculate the differences between the consecutive terms to find a pattern.

1. **Difference between 349 and 6:**
\[ 349 - 6 = 343 \]

2. **Difference between 565 and 349:**
\[ 565 - 349 = 216 \]

3. **Difference between 754 and 565:**
\[ 754 - 565 = 189 \]

4. **Difference between 781 and 754:**
\[ 781 - 754 = 27 \]

### Step 3: List the Differences
Now we have the differences:
- 343
- 216
- ?
- 189
- 27

### Step 4: Identify the Pattern in the Differences
Next, we can see if these differences relate to cubes of integers:
- \(7^3 = 343\)
- \(6^3 = 216\)
- \(5^3 = 125\) (this is the missing difference)
- \(4^3 = 64\)
- \(3^3 = 27\)

### Step 5: Calculate the Missing Number
Since the missing difference corresponds to \(5^3 = 125\), we can find the missing number by adding this difference to the last known number in the series before the missing term.

\[ 565 + 125 = 690 \]

### Step 6: Verify the Pattern
Now, we can check the differences again:
- Difference between 690 and 565:
\[ 690 - 565 = 125 \quad (5^3) \]
- Difference between 754 and 690:
\[ 754 - 690 = 64 \quad (4^3) \]
- Difference between 781 and 754:
\[ 781 - 754 = 27 \quad (3^3) \]

### Conclusion
The missing number in the series is **690**.