The instant you meet a rule of the form un+1=f(un) (or a term built from (−1)n, a derivative, a square root, etc.), write out u1,u2,u3,u4,u5 explicitly. The pattern you need (periodicity, a closed form, a constant ratio) almost always becomes visible in the first handful of terms, and this costs seconds.
Trigger: any sequence given by a recurrence or an index-dependent formula, especially when the question asks for a far-off term or a long sum.
Instances:
(i) an alternating (−1)n rule reveals a short repeating block;
(ii) a rational map un+1=cun+daun+b reveals a cycle;
(iii) a square-root or derivative recurrence reveals exponents in arithmetic progression.
