Number series – fill in the missing term. Sequence: (1/9), (1/3), 1, __, 9, …

Introduction / Context:
This is a classic geometric progression recognition task common in quantitative reasoning tests. We are asked to determine the missing term in a sequence that appears to grow by a consistent multiplicative factor. Identifying the constant multiplier is the key.

Given Data / Assumptions:

• Sequence: (1/9), (1/3), 1, __, 9, …
• Exactly one value is missing.
• Numbers are positive and appear to increase at a steady multiplicative rate.

Concept / Approach:

Check the ratio between consecutive known terms. If the same ratio repeats, the sequence is geometric. Then apply that ratio to leap from one term to the next, including across the blank position.

Step-by-Step Solution:

Verification / Alternative check:

Once we confirm two consecutive ratios are equal (both equal to 3), the geometric nature is established. Extending forward preserves consistency with the given 9, validating the result.

Why Other Options Are Wrong:

(2/3) breaks the factor-of-3 rule. 6 and 27 overshoot because they correspond to multiplying by 6 or 9 rather than 3 at that position, causing inconsistency with the next term 9.

Common Pitfalls:

Misreading a geometric progression as arithmetic (adding instead of multiplying) frequently leads to incorrect results. Always test ratios when values scale quickly.

Final Answer:

3