Fill in the blank: 1, 2, 9, ___, 16900

Concept

In a fast-growing number series like this one, each new term can be built from the two terms immediately before it by a fixed rule. Here, every term (from the third onward) equals the square of the sum of its two preceding terms: an = (an-2 + an-1)2.

Application

1. Given a1 = 1 and a2 = 2.

2. a3 = (a1 + a2)2 = (1 + 2)2 = 32 = 9 — this matches the third given term, confirming the rule is right.

3. a4 = (a2 + a3)2 = (2 + 9)2 = 112 = 121 — this is the missing term.

4. a5 = (a3 + a4)2 = (9 + 121)2 = 1302 = 16900 — this matches the fifth given term exactly.

Cross-check

Working backward from the last given term, the square root of 16900 is 130, and 9 + 121 does equal 130 — an independent confirmation, obtained the opposite way, that the missing term found above is correct.