Find the next term of the following series :
0, 5, 22, 57, 116, ?

Question

Find the next term of the following series : 
0, 5, 22, 57, 116, ?

Accepted Answer

To determine the next term in the series 0, 5, 22, 57, 116, we need to identify the pattern or rule governing the sequence. Let's analyze the differences between consecutive terms:

1. Difference between 5 and 0: $5 - 0 = 5$
2. Difference between 22 and 5: $22 - 5 = 17$
3. Difference between 57 and 22: $57 - 22 = 35$
4. Difference between 116 and 57: $116 - 57 = 59$

Now, let's examine the differences between these differences:

1. Difference between 17 and 5: $17 - 5 = 12$
2. Difference between 35 and 17: $35 - 17 = 18$
3. Difference between 59 and 35: $59 - 35 = 24$

We observe that the second-level differences (12, 18, 24) increase by 6 each time. This suggests that the differences between consecutive terms in the original series are increasing in a quadratic manner.

To find the next difference after 59, we add 6 to the last second-level difference:

$$24 + 6 = 30$$

Adding this to the last first-level difference:

$$59 + 30 = 89$$

Now, add this difference to the last term in the series to find the next term:

$$116 + 89 = 205$$

Thus, the next term in the series is 205, which corresponds to Option 1.

**Explanation for Incorrect Options:**

- **Option 2 (216):** This option does not fit the pattern of differences increasing by 6 at the second level. Using the established pattern, the calculated next term is 205, not 216.

- **Option 3 (192):** This option is also incorrect because it does not align with the calculated difference of 89 added to the last term, which results in 205.

- **Option 4 (207):** Although close, this option does not match the precise calculation of 205 based on the consistent pattern of differences.

Therefore, based on the logical analysis of the series, Option 1 (205) is indeed the correct answer.