A number series is given with one term missing. Observe the sequence 1, 2, 4, 7, ?, 16 and choose the correct alternative that completes the series.

Introduction / Context:
This problem presents a number series with gradually increasing terms. The given sequence is 1, 2, 4, 7, ?, 16. Our job is to determine the missing middle term by uncovering the pattern that describes how each term is generated from the previous one. Many such series are based on increasing differences that follow a simple rule.

Given Data / Assumptions:
Series: 1, 2, 4, 7, ?, 16.
One term is missing between 7 and 16.
The pattern likely uses increasing differences rather than a constant increment.
Exactly one option will make the differences follow a simple sequence.

Concept / Approach:
To solve this, we analyse the differences between consecutive terms. If the differences themselves follow a simple pattern, such as consecutive integers (1, 2, 3, 4, 5, ...), we can extend that pattern to identify the missing term. This type of structure is very common in exam questions and is an important pattern to recognise quickly.

Step-by-Step Solution:
Compute differences between known terms. From 1 to 2: difference = 1. From 2 to 4: difference = 2. From 4 to 7: difference = 3. The differences so far are 1, 2, and 3, which are consecutive integers. Next difference should be 4, so the missing term is 7 + 4 = 11. After that, the difference should be 5, giving 11 + 5 = 16, which matches the last term. Hence, the missing term is 11.

Verification / Alternative check:
With the missing term filled in, the series becomes 1, 2, 4, 7, 11, 16. The differences are 1, 2, 3, 4, 5, which form a perfect sequence of consecutive natural numbers. This is a very standard pattern and provides strong confirmation that 11 is the correct answer.

Why Other Options Are Wrong:
Option A: 14 would give a difference of 7 from 7 and only 2 to 16, which does not follow the consecutive differences pattern.
Option B: 8 gives a difference of 1 from 7 and then 8 to 16, destroying the 1, 2, 3, 4, 5 structure.
Option D: 15 results in differences 8 and 1, which are highly irregular.
Option E: 9 gives differences of 2 and 7, again inconsistent with the simple gradual increase in differences.

Common Pitfalls:
Some candidates may guess based on the approximate size of the missing term rather than systematically working out the differences. Others may try to fit a multiplicative rule where none is needed. In many reasoning questions, increasing difference patterns like 1, 2, 3, 4, and so on appear very frequently. Recognising them quickly can save time and improve accuracy.

Final Answer:
The number that maintains the sequence of differences as 1, 2, 3, 4, 5 is 11, so the correct option is 11.