A series is given with one term missing. Select the correct alternative from the given ones that will complete the series: 2, 7, 16, ?, 46, 67.

Introduction / Context:
This number series increases with steps that become larger each time. The differences between consecutive terms themselves grow linearly. The question tests whether you can identify this sequence of differences and extend it to locate the missing term in the middle of the series.

Given Data / Assumptions:

• Series: 2, 7, 16, ?, 46, 67.
• One middle term is missing.
• The increments between terms increase steadily.

Concept / Approach:
We first compute the differences between the known consecutive terms. Then we examine whether these differences form an arithmetic progression. If they do, we can insert the appropriate missing difference and reconstruct the missing term. This process uses the idea that the main sequence is built from a simpler sequence of step sizes.

Step-by-Step Solution:
Step 1: Compute known differences where possible. 7 - 2 = 5. 16 - 7 = 9. 46 - ? is unknown. 67 - 46 = 21. Step 2: Look at the pattern of known differences: 5, 9, ?, 21. Step 3: Notice that 5, 9, 13, 17, 21 would form an arithmetic progression with common difference 4. Step 4: Therefore, the missing differences are 13 and 17 between the central terms. Step 5: Use 13 as the next difference after 16. The missing term = 16 + 13 = 29. Step 6: Check the following step with 17: 29 + 17 = 46, which matches the given value.

Verification / Alternative check:
With the series completed as 2, 7, 16, 29, 46, 67, recalculate the differences: 7 - 2 = 5, 16 - 7 = 9, 29 - 16 = 13, 46 - 29 = 17, 67 - 46 = 21. These differences are 5, 9, 13, 17, 21, forming an arithmetic progression with common difference 4. This double check shows that the sequence of differences is perfectly regular and validates 29 as the correct missing term.

Why Other Options Are Wrong:
If we try 26, 27, or 31 as the missing term, the resulting differences fail to form the neat sequence 5, 9, 13, 17, 21. For example, if the missing term were 27, then 27 - 16 = 11 and 46 - 27 = 19, giving differences 5, 9, 11, 19, 21, which are irregular. Because the series is clearly built on a difference pattern with constant increment 4, any number that breaks this pattern cannot be correct.

Common Pitfalls:
Some learners look only at the main terms and guess based on rough spacing. Others notice that the differences are increasing but do not check whether they increase in a constant manner. The safest route is always to calculate all possible differences and then inspect them carefully for simple arithmetic sequences.

Final Answer:
The missing number in the series is 29.