What should come in place of question mark (?) in the following number series ?
2 , 5 , 1 6, 57, 240 ?

Options

A. A) 1215
B. B) 1205
C. C) 1225
D. D) 1195

Answer

To find the missing number in the series: 2, 5, 16, 57, 240, ?, we will analyze the pattern in the series step by step.

### Step 1: Identify the pattern in the series
We will look at the differences between consecutive terms to identify a pattern.

- From 2 to 5:
- Difference = 5 - 2 = 3
- From 5 to 16:
- Difference = 16 - 5 = 11
- From 16 to 57:
- Difference = 57 - 16 = 41
- From 57 to 240:
- Difference = 240 - 57 = 183

### Step 2: Analyze the differences
Now, we will analyze the differences we calculated:
- The differences are: 3, 11, 41, 183.

### Step 3: Find the pattern in the differences
Next, we will look at the differences between these differences:
- From 3 to 11:
- Difference = 11 - 3 = 8
- From 11 to 41:
- Difference = 41 - 11 = 30
- From 41 to 183:
- Difference = 183 - 41 = 142

### Step 4: Analyze the second differences
Now we will analyze these second differences:
- The second differences are: 8, 30, 142.

### Step 5: Find the pattern in the second differences
Next, we will look at the differences between these second differences:
- From 8 to 30:
- Difference = 30 - 8 = 22
- From 30 to 142:
- Difference = 142 - 30 = 112

### Step 6: Identify the next term in the series
It appears that the differences are increasing significantly. If we follow the pattern, we can predict the next difference after 183.

Assuming the next difference follows a similar pattern, we can estimate:
- If we add the last difference (183) to the next calculated difference (which we can assume to be a continuation of the pattern), we can find the next term.

### Step 7: Calculate the next term
To find the next term in the series, we can multiply the last term (240) by a factor and add a constant:
- If we assume the next factor is 5 (following the previous multipliers of 2, 3, 4), we can calculate:
- Next term = 240 * 5 + (3 * 5)
- Next term = 1200 + 15 = 1215

### Conclusion
Thus, the missing number in the series is **1215**.