What will come in place of question mark (? ) in the following number series ?
`210 " "209 " "213 " "186" " 202" "?`

Options

A. 138
B. 77
C. 177
D. 327

Answer

To solve the number series: `210, ?, 209, ?, 213, ?, 186, ?, 202, ?`, we need to identify the pattern in the differences between the numbers.

### Step-by-Step Solution:

1. **Identify the Series**: The series is `210, ?, 209, ?, 213, ?, 186, ?, 202, ?`.

2. **Calculate Differences**:
- Let's look at the differences between the numbers we have:
- From 210 to 209: **Difference = -1**
- From 209 to 213: **Difference = +4**
- From 213 to 186: **Difference = -27**
- From 186 to 202: **Difference = +16**

3. **Analyze the Differences**:
- The differences we have so far are:
- -1, +4, -27, +16
- Notice that the differences alternate between negative and positive.

4. **Identify the Pattern**:
- The differences seem to follow a pattern of squares and cubes:
- -1 = (-1)^3
- +4 = (2)^2
- -27 = (-3)^3
- +16 = (4)^2
- This suggests that the next difference should be (-5)^3 = -125.

5. **Calculate the Missing Number**:
- The last known number before the question mark is 202.
- To find the number that comes before 202, we subtract 125:
- 202 - 125 = 77.
- Therefore, the missing number (the one that replaces the question mark) is **77**.

### Conclusion:
The number that comes in place of the question mark is **77**.