The number that comes next in the sequence 0,6, 24, 60, 120, 210 is

Options

A. 240
B. 290
C. 336
D. 324

Answer

To find the next number in the sequence 0, 6, 24, 60, 120, 210, we can analyze the pattern in the sequence step by step.

### Step 1: Identify the sequence
The given sequence is:
0, 6, 24, 60, 120, 210

### Step 2: Look for a pattern
We can observe that the numbers do not follow a simple arithmetic or geometric progression. Therefore, we will explore a different approach.

### Step 3: Analyze the terms
Let's analyze each term:
- The first term is \(0\).
- The second term is \(6\).
- The third term is \(24\).
- The fourth term is \(60\).
- The fifth term is \(120\).
- The sixth term is \(210\).

### Step 4: Check for a cubic pattern
We notice that each term can be expressed in terms of cubes of integers:
- \(1^3 - 1 = 0\)
- \(2^3 - 2 = 6\)
- \(3^3 - 3 = 24\)
- \(4^3 - 4 = 60\)
- \(5^3 - 5 = 120\)
- \(6^3 - 6 = 210\)

### Step 5: Predict the next term
Following the identified pattern, the next term can be calculated as:
- For \(n = 7\):
\[ 7^3 - 7 = 343 - 7 = 336 \]

### Conclusion
Thus, the next term in the sequence is **336**.

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