The original worked lessons remain the main explanations. Use these focused reminders to check the conditions before you start.
Trigonometric terms in a geometric progression
Use the GP product condition with recurrence checks for zero terms.
Retain both ratio signs when the question permits them.
Apply positivity, angle intervals and convergence separately.
For nonzero consecutive terms \(u,v,w\), \(v^2=uw\) gives the common-ratio test. When zeros are allowed, check \(v=ru\) and \(w=rv\) for a finite real \(r\). This package allows \(r=0\) unless a question says otherwise. The product test alone can accept a false candidate such as \((0,0,8)\).