Conditions on a geometric progression routinely produce a quadratic or higher polynomial with multiple roots. Never accept a root blindly: a spurious root usually appears alongside the genuine one. Read the question's constraints and keep only the root that survives them.
Trigger: you reach a polynomial in the ratio r or an unknown parameter p, and the stem mentions convergence, positive terms, or a specific sign.
Instances:
(i) convergence of an infinite sum requires ∣r∣<1, rejecting any root ≥1 or ≤−1;
(ii) a condition that all terms are positive rejects a root that makes the first term negative or forces a negative ratio;
(iii) a term stated as 'real and positive' uniquely selects the positive root of an even power.
