Number Series Pattern Analysis

The question asks to identify the missing number in the sequence: 13, 25, 49, 97, ?

We need to find the pattern governing this number series.

Identifying the Series Pattern

Let's examine the differences between consecutive terms:

• $25 - 13 = 12$
• $49 - 25 = 24$
• $97 - 49 = 48$

The differences are 12, 24, and 48. Notice that each difference is double the previous difference:

• $12 \times 2 = 24$
• $24 \times 2 = 48$

Following this pattern, the next difference should be $48 \times 2 = 96$.

Calculating the Missing Number

To find the missing number, we add this next difference (96) to the last known term (97):

• $97 + 96 = 193$

Alternatively, we can observe another pattern: each term is twice the previous term minus 1.

• $13 \times 2 - 1 = 26 - 1 = 25$
• $25 \times 2 - 1 = 50 - 1 = 49$
• $49 \times 2 - 1 = 98 - 1 = 97$

Applying this pattern to find the next term:

• $97 \times 2 - 1 = 194 - 1 = 193$

Both methods yield the same result.

Conclusion

The number that replaces the question mark (?) is 193.

This corresponds to Option D.