5, 11, 25, 55, 117, 243, ….

Options

A. 511
B. 499
C. 498
D. 497

Explanation

To solve the series 5, 11, 25, 55, 117, 243, we need to identify the pattern in the sequence. Let's analyze the differences between consecutive terms:

1. **Identify the terms:**
- First term (a1) = 5
- Second term (a2) = 11
- Third term (a3) = 25
- Fourth term (a4) = 55
- Fifth term (a5) = 117
- Sixth term (a6) = 243

2. **Calculate the differences between consecutive terms:**
- a2 - a1 = 11 - 5 = 6
- a3 - a2 = 25 - 11 = 14
- a4 - a3 = 55 - 25 = 30
- a5 - a4 = 117 - 55 = 62
- a6 - a5 = 243 - 117 = 126

So, the differences are:
- 6, 14, 30, 62, 126

3. **Calculate the differences of the differences:**
- 14 - 6 = 8
- 30 - 14 = 16
- 62 - 30 = 32
- 126 - 62 = 64

The second differences are:
- 8, 16, 32, 64

4. **Calculate the differences of the second differences:**
- 16 - 8 = 8
- 32 - 16 = 16
- 64 - 32 = 32

The third differences are:
- 8, 16, 32

5. **Calculate the differences of the third differences:**
- 16 - 8 = 8
- 32 - 16 = 16

The fourth differences are:
- 8, 16

6. **Calculate the differences of the fourth differences:**
- 16 - 8 = 8

The fifth difference is constant:
- 8

7. **Recognizing the pattern:**
The pattern in the differences suggests that the series is generated by a polynomial function. The differences show a consistent doubling pattern, indicating that the next difference should be 8 more than the last difference of 126.

8. **Finding the next term:**
The next difference after 126 would be:
- 126 + 8 = 254

Therefore, the next term in the series would be:
- 243 + 254 = 497

So, the next term in the series is **497**.