Key manipulations: (1) Index shift: sum(f(k), k=1 to n) = sum(f(k+1), k=0 to n-1). (2) Splitting: sum(f(k), k=1 to n) = sum(f(k), k=1 to m) + sum(f(k), k=m+1 to n). (3) Reversing: sum(f(k), k=1 to n) = sum(f(n+1-k), k=1 to n). Reversal is particularly useful for symmetric sums. (4) Double sums: $sum_i$ $sum_j$ can sometimes be simplified by changing order of summation.