Find the missing number in the series 1, 6, 15, ?, 45, 66, 91.

Introduction / Context:
This is a classic number series question where each term increases by a changing amount. The sequence of differences follows a simple arithmetic pattern, and recognising this pattern allows us to fill in the missing value.

Given Data / Assumptions:

• Series: 1, 6, 15, ?, 45, 66, 91.
• Exactly one term in the middle is missing.
• The rule governing the series is the same throughout.

Concept / Approach:
The first step is to compute differences between known neighbouring terms. In many aptitude series, these differences form a simple arithmetic progression, even when the original terms do not. Once that progression is identified, the missing term can be reconstructed.

Step-by-Step Solution:
Step 1: Compute some known differences using terms with no gap at both ends: 66 − 45 = 21 and 91 − 66 = 25. Step 2: Also compute the first two differences at the start: 6 − 1 = 5 and 15 − 6 = 9. Step 3: Consider the sequence of differences we know or expect: 5, 9, ?, 21, 25. Step 4: The pattern suggests an arithmetic progression of differences increasing by 4 each time: 5, 9, 13, 17, 21, 25. Step 5: So the missing difference between the third and fourth terms should be 13, and the next difference should be 17 between the fourth and fifth terms. Step 6: From the third term 15, add 13 to get the fourth term: 15 + 13 = 28. Step 7: Check that from this fourth term we can reach 45 with a difference of 17: 28 + 17 = 45, which is already in the series.

Verification / Alternative check:
With the missing number set to 28, the complete series is 1, 6, 15, 28, 45, 66, 91. The differences are 5, 9, 13, 17, 21, 25 which clearly form an arithmetic progression with common difference 4. This confirms that the pattern is correct and that 28 is the only value that fits consistently.

Why Other Options Are Wrong:
If we choose 24, 26, or 32, the resulting differences fail to follow the steady increase of 4. For example, with 24, the difference from 15 is 9 and from 24 to 45 is 21, which breaks the smooth 5, 9, 13, 17, 21, 25 sequence. Therefore these options cannot be correct.

Common Pitfalls:
Students sometimes try to relate each term directly to the previous one through multiplication or squaring, which complicates what is actually a simple pattern in differences. Another pitfall is to look only at part of the series and overlook the later differences, which contain strong clues. Always consider all available terms when extracting the pattern.

Final Answer:
The missing number that preserves the increasing difference pattern is 28.