Standard sums (memorize these):
| Series | Formula |
|---|---|
| sum_{k=1}^{n} k | n(n+1)/2 |
| sum_{k=1}^{n} k^2 | n(n+1)(2n+1)/6 |
| sum_{k=1}^{n} k^3 | [n(n+1)/2]^2 |
| sum_{k=1}^{n} k^4 | n(n+1)(2n+1)(3n^2+3n-1)/30 |
| sum_{k=1}^{n} (2k-1) | n^2 |
| sum_{k=1}^{n} 2k | n(n+1) |
| sum_{k=1}^{n} k(k+1) | n(n+1)(n+2)/3 |
| sum_{k=1}^{n} k(k+1)(k+2) | n(n+1)(n+2)(n+3)/4 |
Beautiful identity: sum(k^3) = [sum(k)]^2. The sum of cubes equals the square of the sum of natural numbers.
General pattern: sum of k(k+1)(k+2)...(k+m-1) = n(n+1)(n+2)...(n+m) / (m+1). This generalizes the first few formulas.