Find the missing term in the following numbers series questons.
800,770,728,672,?,510,

Options

A. 616
B. 600
C. 580
D. 624

Answer

To find the missing term in the series 800, 770, 728, 672, ?, 510, we will analyze the differences between the consecutive terms and identify a pattern.

### Step-by-Step Solution:

1. **Identify the terms of the series:**
- The series is: 800, 770, 728, 672, ?, 510.

2. **Calculate the differences between consecutive terms:**
- Difference between 800 and 770:
\[ 800 - 770 = 30 \]
- Difference between 770 and 728:
\[ 770 - 728 = 42 \]
- Difference between 728 and 672:
\[ 728 - 672 = 56 \]

3. **List the differences:**
- The differences are: 30, 42, 56.

4. **Calculate the second differences:**
- Difference between 30 and 42:
\[ 42 - 30 = 12 \]
- Difference between 42 and 56:
\[ 56 - 42 = 14 \]

5. **List the second differences:**
- The second differences are: 12, 14.

6. **Identify the pattern in the second differences:**
- The second differences are increasing by 2 (12, 14).
- We can predict the next second difference will be:
\[ 14 + 2 = 16 \]

7. **Calculate the next first difference:**
- Adding this to the last first difference (56):
\[ 56 + 16 = 72 \]

8. **Find the missing term:**
- Subtract this difference from the last known term (672):
\[ 672 - 72 = 600 \]

9. **Verify the last difference:**
- Finally, check the difference between the missing term (600) and the last term (510):
\[ 600 - 510 = 90 \]

10. **List all differences for confirmation:**
- The complete list of differences is: 30, 42, 56, 72, 90.
- The second differences are: 12, 14, 16, 18 (confirming our pattern).

### Conclusion:
The missing term in the series is **600**.