लुप्त पद ज्ञात करें : `0,6,24,60,120,210,?`

Options

A. 240
B. 290
C. 336
D. 504

Answer

To find the missing term in the sequence `0, 6, 24, 60, 120, 210, ?`, we can analyze the pattern of the given numbers step by step.

### Step 1: Identify the pattern
Let's list the terms and see if we can find a relationship between them:
- 0
- 6
- 24
- 60
- 120
- 210

### Step 2: Calculate the differences between consecutive terms
We will find the differences between each pair of consecutive terms:
- 6 - 0 = 6
- 24 - 6 = 18
- 60 - 24 = 36
- 120 - 60 = 60
- 210 - 120 = 90

This gives us the first differences:
- 6, 18, 36, 60, 90

### Step 3: Calculate the second differences
Now, let's calculate the differences of the first differences:
- 18 - 6 = 12
- 36 - 18 = 18
- 60 - 36 = 24
- 90 - 60 = 30

This gives us the second differences:
- 12, 18, 24, 30

### Step 4: Calculate the third differences
Next, we calculate the differences of the second differences:
- 18 - 12 = 6
- 24 - 18 = 6
- 30 - 24 = 6

The third differences are constant:
- 6, 6, 6

### Step 5: Predict the next term
Since the third differences are constant, we can predict the next second difference by adding 6 to the last second difference:
- Next second difference = 30 + 6 = 36

Now, we can find the next first difference by adding this new second difference to the last first difference:
- Next first difference = 90 + 36 = 126

Finally, we can find the next term in the sequence by adding this new first difference to the last term:
- Next term = 210 + 126 = 336

### Conclusion
The missing term in the sequence is **336**.