A number series is given with one term missing. Carefully examine the pattern in 5, 10, 13, 26, 29, 58, 61, ? and choose the correct alternative from the options that will complete the series.

Introduction / Context:
This is a slightly more involved number series problem. The given sequence is 5, 10, 13, 26, 29, 58, 61, ?. The pattern is not a simple arithmetic or geometric progression. Instead, it alternates between two different operations. Such designs are common in exam questions where the examiner wants to test careful observation and the ability to detect a repeating cycle of operations.

Given Data / Assumptions:
Series: 5, 10, 13, 26, 29, 58, 61, ?
Exactly one term is missing at the end of the series.
The rule may involve alternating operations such as multiplication and addition.
A single, consistent pattern should explain all transitions between terms.

Concept / Approach:
Looking at the series, we can try grouping the transitions in pairs. From 5 to 10 and 10 to 13, from 13 to 26 and 26 to 29, and so on. This suggests that one step may involve multiplication and the next a smaller addition. We look for a simple relationship like "multiply by 2, then add 3" repeated again and again. Such alternating two-step patterns are very common in number series questions.

Step-by-Step Solution:
From 5 to 10: 5 * 2 = 10. From 10 to 13: 10 + 3 = 13. From 13 to 26: 13 * 2 = 26. From 26 to 29: 26 + 3 = 29. From 29 to 58: 29 * 2 = 58. From 58 to 61: 58 + 3 = 61. Following the same pattern, the next step should be multiplication by 2 again. So, 61 * 2 = 122. Therefore, the missing term is 122.

Verification / Alternative check:
We verify by summarising the rule: multiply by 2, then add 3, and repeat. Applying this cyclic rule from the first term generates the entire series without any inconsistency. The final step, 61 * 2 = 122, continues the cycle perfectly. None of the other options maintain this simple and elegant pattern, which confirms that 122 is the correct answer.

Why Other Options Are Wrong:
Option B: 125 would require adding 64 instead of following the multiplication by 2 step, which is not supported by the previous terms.
Option C: 128 is 2^7 and does not arise naturally from 61 under the observed rule of multiply by 2 then add 3 in alternation.
Option D: 64 would correspond to adding only 3 to 61, while the pattern demands a multiplication at this stage.
Option E: 118 does not fit either multiply by 2 or add 3, so it breaks the cycle.

Common Pitfalls:
Many candidates focus only on differences: 5, 3, 13, 3, 29, 3, which looks uneven and confusing. They may ignore the fact that some jumps are too large for simple addition and thus indicate multiplication. Recognising alternating operations is crucial. Another common mistake is to search for one single arithmetic progression instead of a repeated two-step pattern.

Final Answer:
The term that correctly completes the cycle of multiply by 2 and then add 3 is 122, so the correct option is 122.