Find the next number in the given number sequence: 4, 15, 37, 81, ?

Introduction / Context:
This question tests recognition of a pattern where each term is obtained from the previous one by doubling and then adding a changing value. Many exam number series rely on such multiply and add structures where the additive part itself follows a simple pattern. Here, we must extend the sequence 4, 15, 37, 81 by one more term and identify it from the options.

Given Data / Assumptions:
- Given sequence: 4, 15, 37, 81, ?- Only the last term is missing.- The series is strictly increasing.- The pattern is likely to involve multiplication and addition rather than constant differences, because the gaps between numbers grow quickly.

Concept / Approach:
When the gaps between terms increase rapidly, it is natural to look for a rule of the form: next term = previous term * k + c, where k and c vary in a simple pattern. Checking the relationship between consecutive terms by reversing this operation often reveals a clear sequence of additive constants that helps to generalise the rule for the next step.

Step-by-Step Solution:
- From 4 to 15: 4 * 2 + 7 = 8 + 7 = 15.- From 15 to 37: 15 * 2 + 7 = 30 + 7 = 37.- From 37 to 81: 37 * 2 + 7 = 74 + 7 = 81.- We see a stable pattern: each term is twice the previous term plus 7.- To find the next term, apply the same rule to 81.- Calculation: 81 * 2 + 7 = 162 + 7 = 169.- So, the missing number in the series is 169.

Verification / Alternative check:
- We can confirm by recalculating each step: 4 → 15, 15 → 37, 37 → 81, and 81 → 169 all follow the rule previous term * 2 + 7.- The sequence of differences (11, 22, 44, 88) also shows a doubling pattern, which is consistent with the multiply by two structure.- No inconsistency appears when 169 is added, showing that the rule holds throughout.

Why Other Options Are Wrong:
- 196, 144, 121 and 225 do not satisfy the rule when placed as the fifth term; they cannot be written as 81 * 2 + 7.- Using these alternatives would break either the doubling pattern or the constant addition of 7.

Common Pitfalls:
- Candidates sometimes try to fit a pattern to the differences alone and may miss the more direct multiply and add relationship.- Another pitfall is assuming a quadratic formula or more complex function when a simple two step rule is sufficient.- Ignoring the repeated appearance of plus 7 and focusing only on the rapid growth of the numbers can also lead to confusion.

Final Answer:
Applying the clear rule next term = previous term * 2 + 7 to 81 gives 81 * 2 + 7 = 169, so 169 is the correct next number in the series.