In the following question, a number series is given with one term missing. Select the missing number from the series that correctly continues the pattern: 2, 5, 17, 71, ?

Introduction / Context:
This series 2, 5, 17, 71, ? is more challenging than simple arithmetic or geometric series. The jumps become increasingly large, which often indicates a pattern involving multiplication by increasing numbers along with additional constants. Such series test advanced pattern recognition skills in reasoning exams.

Given Data / Assumptions:

• Series: 2, 5, 17, 71, ?
• Options: 131, 247, 359, 419.
• We assume a consistent rule linking each term to its predecessor.

Concept / Approach:
The key idea is to see whether we can express each term as the previous term multiplied by an increasing integer, plus or minus a constant that also follows a pattern. Many known solutions for this question use a relation of the form T(n+1) = T(n) * k + c, where k increases and c has its own logic. We test small integer multipliers and see what constant is required each time, then continue the pattern to find the missing term.

Step-by-Step Solution:
Step 1: Express 5 in terms of 2.5 can be written as 2 * 2 + 1.Step 2: Express 17 in terms of 5.17 can be written as 5 * 3 + 2.Step 3: Express 71 in terms of 17.71 can be written as 17 * 4 + 3.Step 4: Observe the pattern.Multipliers are 2, 3, 4, which are consecutive integers.Added constants are 1, 2, 3, which also increase by 1 each time.Step 5: Apply the same logic to find the next term.Next multiplier should be 5, and the next constant should be 4.Missing term = 71 * 5 + 4 = 355 + 4 = 359.

Verification / Alternative check:
Write the completed series with its generation rule: 2, 2 * 2 + 1 = 5, 5 * 3 + 2 = 17, 17 * 4 + 3 = 71, 71 * 5 + 4 = 359. Both the multiplier sequence (2, 3, 4, 5) and the addend sequence (1, 2, 3, 4) are neat and consistent. No other option can preserve such a clean structure throughout the series.

Why Other Options Are Wrong:
131: Cannot be expressed as 71 * 5 plus a constant that continues the 1, 2, 3 pattern.247: Also fails to fit the simple rule of multiplier 5 and addend 4.419: Would require adding a much larger constant that does not follow the slow incremental pattern 1, 2, 3, 4.

Common Pitfalls:
Some learners only search for patterns in pure differences between consecutive terms, which become 3, 12, 54 here and may appear unmanageable. However, expressing each term as a product of the previous term with a small integer plus a steadily increasing constant is often easier and more natural. Always try simple multiplier plus addend patterns before giving up on a complex looking series.

Final Answer:
The missing number that continues the pattern T(n+1) = T(n) * k + c with k and c increasing by 1 is 359.