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

Options

A. 205
B. 216
C. 192
D. 207

Answer

To find the next term in the series: 0, 5, 22, 57, 116, we can analyze the pattern in the series step by step.

### Step 1: Identify the pattern
We observe the given terms:
- 1st term: 0
- 2nd term: 5
- 3rd term: 22
- 4th term: 57
- 5th term: 116

### Step 2: Relate the terms to cubes
We can see that each term can be expressed in relation to cubes of natural numbers:
- 1st term: \(1^3 - 1 = 0\)
- 2nd term: \(2^3 - 3 = 5\)
- 3rd term: \(3^3 - 5 = 22\)
- 4th term: \(4^3 - 7 = 57\)
- 5th term: \(5^3 - 9 = 116\)

### Step 3: Identify the pattern in the subtracted numbers
The numbers being subtracted are: 1, 3, 5, 7, 9. This sequence is an arithmetic series where each term increases by 2. The next number in this sequence would be 11.

### Step 4: Calculate the next term
Now, for the 6th term:
- We take \(6^3\) and subtract 11:
\[ 6^3 = 216 \]
\[ 216 - 11 = 205 \]

### Conclusion
Thus, the next term in the series is **205**.