Technique 1: Direct Formula If general term is a polynomial in k, decompose into sum(k^3), sum(k^2), sum(k), and constants. Example: sum k(k+1) = sum(k^2+k) = n(n+1)(2n+1)/6 + n(n+1)/2.

Technique 2: Telescoping Write t_k = f(k) - f(k+1). Sum collapses to f(1) - f(n+1). Use partial fractions: 1/(k(k+1)) = 1/k - 1/(k+1). For products of 3 terms: 1/(k(k+1)(k+2)) = (1/2)(1/(k(k+1)) - 1/((k+1)(k+2))).

Technique 3: S - rS (AGP) For sum of (AP part)*(GP part): multiply by r, subtract, simplify. The AP component telescopes, leaving a GP.

Technique 4: Method of Differences When differences of terms form AP or GP: compute differences, identify pattern, find general term.

Technique 5: Vn Method (Factorial/Product sums) For sum of kk!: use kk! = (k+1)! - k!. For sum of k(k+1)...(k+m): use the generalized formula.

Technique 6: Rationalization For 1/(sqrt(k)+sqrt(k+1)): multiply by (sqrt(k+1)-sqrt(k)) to telescop.

Choice guide: Polynomial in k -> Technique 1. Fractions with products -> Technique 2. AP*GP -> Technique 3. Unknown pattern -> Technique 4. Square roots -> Technique 6.