600 519 ? 406 370 345 329

Options

A. 435
B. 455
C. 425
D. 445

Explanation

To find the missing number in the series: 600, 519, ?, 406, 370, 345, 329, we can follow these steps:

### Step 1: Identify the series
The series is: 600, 519, ?, 406, 370, 345, 329.

### Step 2: Calculate the differences between consecutive numbers
Let's calculate the differences between the known consecutive numbers in the series.

1. Difference between 600 and 519:
\[ 600 - 519 = 81 \]

2. Difference between 519 and the missing number (let's call it \( x \)):
\[ 519 - x \]

3. Difference between the missing number \( x \) and 406:
\[ x - 406 \]

4. Difference between 406 and 370:
\[ 406 - 370 = 36 \]

5. Difference between 370 and 345:
\[ 370 - 345 = 25 \]

6. Difference between 345 and 329:
\[ 345 - 329 = 16 \]

### Step 3: Analyze the differences
So far, we have the differences:
- 81 (from 600 to 519)
- \( 519 - x \) (from 519 to \( x \))
- \( x - 406 \) (from \( x \) to 406)
- 36 (from 406 to 370)
- 25 (from 370 to 345)
- 16 (from 345 to 329)

### Step 4: Identify a pattern in the differences
The differences we have calculated are:
- 81
- \( 519 - x \)
- \( x - 406 \)
- 36
- 25
- 16

We can see that the differences are decreasing:
- 81
- 36
- 25
- 16

### Step 5: Establish a pattern
The differences seem to follow a pattern of squares:
- \( 9^2 = 81 \)
- \( 6^2 = 36 \)
- \( 5^2 = 25 \)
- \( 4^2 = 16 \)

The missing difference should logically be \( 7^2 = 49 \).

### Step 6: Set up the equation
From the pattern, we can set up the following equations:
1. \( 519 - x = 49 \)
2. \( x - 406 = 36 \)

### Step 7: Solve for \( x \)
From the first equation:
\[ x = 519 - 49 = 470 \]

Now, let's check if this value satisfies the second equation:
\[ 470 - 406 = 64 \]
This does not match our earlier calculation of 36, indicating that we need to re-evaluate our approach.

### Step 8: Correct the missing number
Instead, let's check the next difference:
If we assume the missing number is \( 455 \):
1. Check \( 519 - 455 = 64 \)
2. Check \( 455 - 406 = 49 \)

Both calculations yield valid differences.

### Conclusion
The missing number in the series is:
\[ \boxed{455} \]