In the following question, select the missing number from the given series: 3, 7, 16, 35, ?

Introduction / Context:
This series involves numbers that increase more rapidly than a simple arithmetic progression. The differences between consecutive terms grow in a structured way, and second differences reveal the true pattern. Recognising how second differences behave is a key reasoning skill for solving advanced number series questions of this type.

Given Data / Assumptions:

• Series: 3, 7, 16, 35, ?
• Exactly one term is missing at the end.
• The increases between terms are not constant.

Concept / Approach:
We first compute the differences between consecutive terms and then compute the differences between those differences (second differences). If the second differences follow a simple pattern, for example doubling each time, we can extend this pattern to obtain the next first difference and hence the next term in the original series. This is essentially using the idea of a quadratic like growth in the sequence.

Step-by-Step Solution:
Step 1: Compute the first differences. 7 - 3 = 4. 16 - 7 = 9. 35 - 16 = 19. Step 2: Collect these: 4, 9, 19. Step 3: Compute the second differences (differences of the first differences). 9 - 4 = 5. 19 - 9 = 10. Step 4: The second differences 5 and 10 show a doubling pattern: each step multiplies by 2. Step 5: Continuing this, the next second difference should be 20. Step 6: Add this to the last first difference: 19 + 20 = 39. Step 7: Now add this new first difference to the last term of the original series: 35 + 39 = 74.

Verification / Alternative check:
Rewrite the series including the missing term: 3, 7, 16, 35, 74. The first differences are 4, 9, 19, 39. The second differences are 9 - 4 = 5, 19 - 9 = 10, 39 - 19 = 20. These second differences clearly double each time, forming 5, 10, 20, which is a consistent and simple rule. This confirms that 74 is the correct next term maintaining the structure of the series.

Why Other Options Are Wrong:
If we choose 73, 78, or 82, the first and second differences no longer follow the exact doubling rule. For example, using 78 would give a last first difference of 78 - 35 = 43, and the second differences would not form a neat pattern of 5, 10, 20. Because the given steps clearly indicate a doubling behaviour at the level of second differences, any number that spoils this regularity cannot be the correct answer.

Common Pitfalls:
Some learners stop after checking that the first differences are increasing but do not investigate how they are increasing. Others try to force the numbers into linear or simple multiplicative patterns, which do not fit well. Whenever first differences appear to grow in a non uniform way, always consider second differences as a tool to uncover a hidden but simple rule.

Final Answer:
The missing number that correctly completes the series is 74.