17 18 26 53 117 ? 458

Options

A. 342
B. 142
C. 257
D. 242

Explanation

To solve the series: 17, 18, 26, 53, 117, ?, 458, we will analyze the differences between consecutive terms and identify a pattern.

### Step-by-Step Solution:

1. **Identify the given series:**
The series is: 17, 18, 26, 53, 117, ?, 458

2. **Calculate the differences between consecutive terms:**
- Difference between 18 and 17:
\( 18 - 17 = 1 \)
- Difference between 26 and 18:
\( 26 - 18 = 8 \)
- Difference between 53 and 26:
\( 53 - 26 = 27 \)
- Difference between 117 and 53:
\( 117 - 53 = 64 \)

So, the differences are: 1, 8, 27, 64.

3. **Identify the pattern in the differences:**
The differences can be recognized as cubes of integers:
- \( 1 = 1^3 \)
- \( 8 = 2^3 \)
- \( 27 = 3^3 \)
- \( 64 = 4^3 \)

This suggests that the next difference should be \( 5^3 = 125 \).

4. **Calculate the next term in the series:**
To find the missing term (?), we add the next difference (125) to the last known term (117):
\[ 117 + 125 = 242 \]

5. **Verify the next term with the last term in the series:**
The next difference after 242 should be \( 6^3 = 216 \):
\[ 242 + 216 = 458 \]
This confirms that our calculations are correct.

6. **Final answer:**
The missing term in the series is **242**.