Some Special Sequences — Formula Sheet

Σn = sum of first n natural numbers = n(n+1)/2

Σn² = sum of squares of first n natural numbers = n(n+1)(2n+1)/6

For Σn²

Use the identity n³ - (n-1)³ = 3n² - 3n + 1. Write this for n = 1, 2, 3, ..., n and add.

For Σn³

Use the identity n⁴ - (n-1)⁴ = 4n³ - 6n² + 4n - 1. Write this for n = 1, 2, ..., n and add.

The series 1 + 2 + 3 + ... + n is an arithmetic series with first term a = 1 and common difference d = 1.

Using the AP sum formula

Σn = n(n+1)/2.