SMO Senior · 3. Polynomials, functions and sequences
Collapse long sums by telescoping and pairing, read polynomials through special values and auxiliary polynomials, use Vieta and power-sum recurrences instead of solving, and handle periodic sequences and functional equations.
Evaluate telescoping and paired sums correctly.
Use special values, the remainder and factor theorems and auxiliary polynomials.
Apply Vieta's formulas and power-sum recurrences to symmetric expressions.
Find periods or closed forms of recursive sequences.
Solve functional equations by substitutions and verify the answers.