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

Introduction / Context:
This series is an example of a geometric progression, where each term is obtained by multiplying the previous term by a fixed factor. Identifying that factor makes it straightforward to determine the missing term. Such questions are common in aptitude tests to assess recognition of multiplicative patterns in number sequences.

Given Data / Assumptions:
Given series: 6, 18, 54, ?, 486, 1458.One term, between 54 and 486, is missing.We assume a constant multiplicative factor connects consecutive terms.

Concept / Approach:
In a geometric sequence, each term equals the previous term multiplied by a common ratio. By dividing a term by its predecessor, we can find this ratio. Once the ratio is identified, we apply it consistently to both known and missing positions in the series. The neat exponential growth in the values here strongly indicates a simple integer ratio, especially 3.

Step-by-Step Solution:
Step 1: Compute 18 / 6 = 3, so the second term is 3 times the first term.Step 2: Compute 54 / 18 = 3, so the third term is 3 times the second term.Step 3: The pattern suggests that each term is obtained by multiplying the previous term by 3.Step 4: Let the missing term be x. Then x = 54 * 3 = 162.Step 5: Verify with the next known term: 162 * 3 = 486, which matches the given series.Step 6: Check the last step: 486 * 3 = 1458, confirming that the factor 3 is consistent throughout.

Verification / Alternative check:
Write the completed series: 6, 18, 54, 162, 486, 1458. The sequence clearly shows each term as three times the previous one. This multiplicative pattern is simple, consistent, and matches all known terms, providing strong confirmation that 162 is indeed the correct missing term according to the geometric progression rule.

Why Other Options Are Wrong:
Option 164 does not satisfy 54 * 3 and also fails to produce 486 when multiplied by any simple factor.Option 160 breaks the ratio, since 160 * 3 = 480, not 486, so the pattern would not match the given fifth term.Option 168 would give 168 * 3 = 504, again inconsistent with the given value 486.

Common Pitfalls:
Some test takers try to apply addition instead of multiplication, especially when numbers grow quickly, which leads to inconsistent differences. Another common issue is miscalculating products such as 54 * 3. In exam conditions, double-checking simple multiplications can avoid unnecessary errors in otherwise straightforward questions.

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