Select the option that is related to the fifth number in the same way as the…

Concept: In a number analogy of the form a : b, one single arithmetic rule must carry the first term to the second term in every pair that is supplied. When the supplied second terms are perfect squares, the productive move is to compare each first term with the square root of its partner and to fit a linear rule of the form root = p × a + q. Such a rule is accepted only after it reproduces every supplied pair exactly, and only then may it be extended to the term whose partner is missing.

Application to this analogy:

1. Write each supplied second term as a square: 9 = 32 and 121 = 112, so the two roots are 3 and 11.

2. Set each first term beside its own root: 2 goes with 3, and 6 goes with 11.

3. Fit root = p × a + q to those two points: p = (11 − 3) ÷ (6 − 2) = 8 ÷ 4 = 2, and q = 3 − (2 × 2) = −1, which gives root = 2a − 1.

4. Verify the fitted rule on both supplied pairs: for a = 2, 2a − 1 = 3 and 32 = 9; for a = 6, 2a − 1 = 11 and 112 = 121. Both supplied pairs are reproduced exactly.

5. Extend the verified rule to the fifth number: for a = 5, 2a − 1 = 9 and 92 = 81.

Cross-check: Expanding the rule gives b = (2a − 1)2 = 4a2 − 4a + 1, so the expanded form must return the same three values.

The root form and the expanded form agree on every supplied pair, so under this analogy the fifth number 5 is matched with 81.