In the following question, a number series is given with one missing term. Select the correct number from the given options that will complete the series. 9, 10, 13, 22, 49, ?

Introduction / Context:
This question involves a number series where the gaps between successive terms increase according to a pattern. The learner must study the differences between consecutive terms and identify a rule that can be extended to compute the missing value. Recognising structured growth in the differences is a core skill in number series reasoning.

Given Data / Assumptions:
- Given series: 9, 10, 13, 22, 49, ? - The sixth term is missing. - The pattern is expected to depend on differences between terms.

Concept / Approach:
Many number series use a pattern in the first differences or even second differences. Here, we inspect the differences between consecutive terms. If the differences themselves form a recognisable pattern, such as powers of a number or multiples with a fixed ratio, that pattern can be extended to find the next term in the original series.

Step-by-Step Solution:
Step 1: Compute differences: 10 - 9 = 1, 13 - 10 = 3, 22 - 13 = 9, 49 - 22 = 27. Step 2: Observe that these differences are 1, 3, 9, and 27. Step 3: Notice that 1, 3, 9, 27 are consecutive powers of 3: 3^0 = 1, 3^1 = 3, 3^2 = 9, 3^3 = 27. Step 4: The next difference should naturally continue this pattern as 3^4 = 81. Step 5: Add 81 to the last known term to get the missing value: 49 + 81 = 130.

Verification / Alternative check:
Verify by reconstructing the series from the starting term. Begin with 9, then add 1, 3, 9, 27, and finally 81. This generates 9, 10, 13, 22, 49, 130, which matches the given sequence and confirms that 130 is the correct missing term.

Why Other Options Are Wrong:
- Option 117 implies a final difference of 68, which does not fit the powers of three pattern. - Option 126 gives a last difference of 77, which again does not match any power of three. - Option 115 gives a difference of 66 and breaks the systematic pattern in the consecutive differences.

Common Pitfalls:
Some learners look only for linear patterns, such as constant or steadily increasing differences, and may miss that the differences themselves follow an exponential rule. Others might try to fit an arbitrary formula directly to the terms instead of first analysing the simpler sequence of differences. A clear and methodical approach to differences greatly simplifies such questions.

Final Answer:
The missing term in the series is 130, so the correct option is 130.