Find the next term of the series :
0, 6, 24, 60, 120, ?

Question

Find the next term of the series :
 0, 6, 24, 60, 120, ?

Accepted Answer

To determine the next term in the series 0, 6, 24, 60, 120, we need to identify the pattern or rule governing the sequence. Let's analyze the series step by step:

1. **Identify the pattern:**
   - The series appears to be related to the cubes of natural numbers. Let's examine the differences between consecutive terms:
     - $6 - 0 = 6$
     - $24 - 6 = 18$
     - $60 - 24 = 36$
     - $120 - 60 = 60$

2. **Recognize the pattern in differences:**
   - The differences themselves form a sequence: 6, 18, 36, 60. Notice that these differences are related to the squares of natural numbers multiplied by 3:
     - $6 = 3 \times 1^2$
     - $18 = 3 \times 3^2$
     - $36 = 3 \times 6^2$
     - $60 = 3 \times 10^2$

3. **Predict the next difference:**
   - Following the pattern, the next difference should be $3 \times 15^2 = 90$.

4. **Calculate the next term:**
   - Add this difference to the last term in the series:
     - $120 + 90 = 210$

Thus, the next term in the series is 210, which corresponds to Option 2.

**Explanation for why other options are incorrect:**

- **Option 1 (180):** This option does not fit the pattern of differences derived from the sequence. Adding 180 to the last term (120) would not follow the established rule of multiplying squares by 3.
  
- **Option 3 (216):** Similarly, adding 216 to the last term does not align with the pattern of differences. The calculated difference of 96 (216 - 120) does not fit the sequence of differences derived from the squares multiplied by 3.

- **Option 4 (240):** This option results in a difference of 120 (240 - 120), which again does not match the pattern of differences established by the sequence.

Therefore, based on the logical analysis and pattern recognition, Option 2 (210) is the correct answer.