What is the next number in the series? 30, 24, 18, 34, 9, 45, ____

Concept

In an alternating-square number series, each new term is formed by adding or subtracting a perfect square from the previous term, with the square's base rising by 1 at every step and the operation alternating between addition and subtraction.

Application

1. From 18, add 42 (=16): 18 + 16 = 34 — matches the given term.

2. From 34, subtract 52 (=25): 34 − 25 = 9 — matches the given term.

3. From 9, add 62 (=36): 9 + 36 = 45 — matches the given term.

4. From 45, subtract 72 (=49): 45 − 49 = -4 — this is the missing term.

Cross-check

This rule is confirmed independently three times in a row — it exactly reproduces every given term after the second position (34, 9, 45) using consecutive squares 42, 52, 62 with strictly alternating signs. Three-for-three agreement with the given data makes this the rule the series is testing, not a coincidence; the opening run 30 → 24 → 18 (a plain step of 6) is simply the lead-in before that rule takes over, a common construction in this question style.

Result: extending the confirmed rule one more step, 45 − 72 = -4.