In the number series 9, 18, 72, 576, ?, the terms grow rapidly. By identifying the pattern in the multipliers, choose the correct number to complete the series.

Introduction / Context:
This question presents a rapidly increasing number series, which typically indicates a multiplicative pattern rather than one based on small differences. Your task is to recognize the pattern in the multipliers applied at each step and then extend that pattern to determine the next term in the sequence.

Given Data / Assumptions:
The series is:
9, 18, 72, 576, ?
We assume that each term is generated by multiplying the previous term by a certain factor, and that these factors themselves follow a simple progression, often involving powers of 2 or another small integer.

Concept / Approach:
We start by examining the ratio between consecutive terms in the series. When those ratios show a clear and predictable pattern, such as doubling, tripling, or following powers of a particular number, we can use that pattern to find the missing term. The presence of values like 72 and 576 suggests strong multiplicative behavior.

Step-by-Step Solution:
Step 1: Compute the ratios between consecutive terms.18 / 9 = 272 / 18 = 4576 / 72 = 8Step 2: Observe that the multipliers are 2, 4, and 8, which are powers of 2: 2^1, 2^2, and 2^3.Step 3: The pattern suggests that the next multiplier should be the next power of 2 in this sequence, namely 2^4 = 16.Step 4: Multiply the last known term by 16.576 * 16 = 9216.

Verification / Alternative check:
We can describe the series generation rule as follows: start with 9, then multiply by 2 to get 18, multiply by 4 to get 72, multiply by 8 to get 576, and finally multiply by 16 to obtain the missing term. This yields 9, 18, 72, 576, 9216. The sequence of multipliers is 2, 4, 8, 16, clearly following powers of 2, which confirms that the pattern is correct and the computed term is consistent.

Why Other Options Are Wrong:
The alternative values 8116, 8216, 9016, or 9600 do not equal 576 multiplied by 16 and do not fit any simple extension of the powers of 2 pattern. For instance, if we attempted to use 9016, the ratio 9016 / 576 would not be a power of 2 and would break the elegant structure of the multipliers that define the series.

Common Pitfalls:
Some candidates may mistakenly look at differences between terms, which become very large and irregular in multiplicative series, making that approach ineffective. Others might miscalculate one of the products, especially 576 times 16, which requires careful multiplication. Always confirm the multipliers carefully and verify that they form a simple and consistent progression.

Final Answer:
The next term in the series, obtained by multiplying 576 by 16, is 9216.