A series is given with one number missing. Choose the correct alternative from the given options that will complete the geometric number series: 6, 18, 54, ?, 486, 1458.

Introduction / Context:
This is a geometric progression type number series, where each term is obtained by multiplying the previous term by a fixed factor. You are asked to identify the missing term. Such questions are frequently used to test recognition of multiplicative patterns rather than additive patterns.

Given Data / Assumptions:

- The series is 6, 18, 54, ?, 486, 1458.- One term between 54 and 486 is missing.- The pattern is consistent for the entire sequence.

Concept / Approach:
When numbers grow very quickly, we should suspect multiplication rather than addition. To detect a geometric pattern, divide each term by the previous one and see if the ratio is constant. If a constant ratio exists, the series is a geometric progression, and we can find missing terms by multiplying or dividing by this ratio.

Step-by-Step Solution:
Step 1: Compute the ratio between the second and first terms: 18 / 6 = 3.Step 2: Compute the ratio between the third and second terms: 54 / 18 = 3.Step 3: So the common ratio is 3.Step 4: Apply this ratio to find the missing term after 54: 54 * 3 = 162.Step 5: Verify the next term using the same ratio: 162 * 3 = 486, which matches the series.Step 6: Finally, check the last term: 486 * 3 = 1458, confirming that the ratio 3 holds throughout.

Verification / Alternative check:
After inserting 162, the completed series is 6, 18, 54, 162, 486, 1458. To verify, check each ratio: 18 / 6 = 3, 54 / 18 = 3, 162 / 54 = 3, 486 / 162 = 3 and 1458 / 486 = 3. The consistent ratio shows that this is a perfect geometric progression with common ratio 3, and 162 is therefore correct.

Why Other Options Are Wrong:
Options 160, 164 and 168 are all close to 162 but none of them produce a constant ratio of 3 when used in the series. For example, if we put 160, then 160 * 3 = 480, which does not equal 486. Any small deviation from 162 breaks the clean multiplication pattern of the entire series, so these values cannot be correct.

Common Pitfalls:
Some students try to look at differences instead of ratios in a rapidly increasing series. While difference methods work for arithmetic progressions, they are inefficient for geometric progressions. When numbers are tripling or doubling, always check for a constant ratio first, as that usually reveals the underlying rule immediately.

Final Answer:
The missing term that completes the geometric number series is 162.