Which number should come in place of the underline mark in the series: 3, 9, 21, ___, 93

Introduction / Context:
This question presents a simple but elegant number series where each term is generated by multiplying the previous term and then adding a fixed constant. Such patterns are widely used in aptitude tests because they are straightforward once the core rule is spotted. We must determine which number fills the blank between 21 and 93 in the series 3, 9, 21, ___, 93.

Given Data / Assumptions:
- Sequence: 3, 9, 21, ?, 93.- Exactly one term is missing.- The series is strictly increasing.

Concept / Approach:
A good starting point is to check how each term relates to the previous one. If we can express each new term as previous term multiplied by a constant and then increased by a small number, we might reveal a rule such as a(n+1) = a(n) * k + c. We will test this hypothesis by working from left to right and verifying whether the same rule holds for every known transition.

Step-by-Step Solution:
- From 3 to 9: 3 * 2 + 3 = 6 + 3 = 9.- From 9 to 21: 9 * 2 + 3 = 18 + 3 = 21.- This suggests the rule: next term = previous term * 2 + 3.- Now apply it to find the missing term after 21: 21 * 2 + 3 = 42 + 3 = 45.- Apply it once more to confirm the last term: 45 * 2 + 3 = 90 + 3 = 93.- So the missing number in the series is 45.

Verification / Alternative check:
- Recomputed full series: 3, 9, 21, 45, 93.- Each step follows the same rule: multiply by 2 and then add 3.- No inconsistency appears, confirming 45 as the unique valid candidate.

Why Other Options Are Wrong:
- 39, 48, 51 and 57 do not satisfy the relation a(n+1) = a(n) * 2 + 3 when placed between 21 and 93.- For example, with 39 as the missing term, 39 * 2 + 3 = 81, which does not equal 93.

Common Pitfalls:
- Some candidates focus only on the differences (6, 12, 24, 48) and get lost in their rapid growth, instead of spotting the simpler multiply and add rule.- Another error is assuming a changing or alternating pattern when a single consistent rule actually works for the whole series.- Overcomplicating the relationship by looking for higher powers or factorials is unnecessary here.

Final Answer:
The series follows the rule next term = previous term * 2 + 3, so the missing number is 45.