What will be the next number in the following series?
0, 2, 9, 28, 65, __________

Options

A. 81
B. 126
C. 256
D. 125

Answer

To find the next number in the series 0, 2, 9, 28, 65, we can analyze the pattern in the sequence step by step.

### Step 1: Identify the series
The given series is:
0, 2, 9, 28, 65

### Step 2: Look for a pattern
Let's examine the differences between consecutive terms:
- 2 - 0 = 2
- 9 - 2 = 7
- 28 - 9 = 19
- 65 - 28 = 37

So, the differences are:
2, 7, 19, 37

### Step 3: Find the differences of the differences
Now, let's find the differences of these differences:
- 7 - 2 = 5
- 19 - 7 = 12
- 37 - 19 = 18

So, the second differences are:
5, 12, 18

### Step 4: Find the differences of the second differences
Next, we find the differences of the second differences:
- 12 - 5 = 7
- 18 - 12 = 6

So, the third differences are:
7, 6

### Step 5: Identify a pattern in the differences
The third differences are not constant, but we can see that they are decreasing by 1. This suggests that we might be dealing with a polynomial of degree 3.

### Step 6: Check for cubic numbers
Now, let's check if the terms can be represented as cubes:
- 0 = 1^3 - 1
- 2 = 2^3 - 6
- 9 = 3^3 - 18
- 28 = 4^3 - 36
- 65 = 5^3 - 60

This suggests a pattern related to cubes. Let's check the cubes:
- 1^3 = 1
- 2^3 = 8
- 3^3 = 27
- 4^3 = 64
- 5^3 = 125

### Step 7: Calculate the next term
Following the pattern, the next term should be:
- 6^3 - 6 * 6 = 216 - 6 * 6 = 216 - 36 = 180

However, we need to calculate the next term based on the previous observations:
If we follow the cube pattern:
- 6^3 = 216

### Step 8: Determine the correct answer
The next number in the series is:
126 (as observed from the pattern of cubes).

Thus, the next number in the series is **126**.