3.1 + 3.2: Sums of natural numbers, squares and cubes

Formula to find the sum of a series of constant terms:

ⁿ∑_r=1 1 = n

Formula to find the sum of the series that does not start at 1:

ⁿ∑_r=k f(r) = ⁿ∑_r=1 f(r) - ^k-1∑_r=1 f(r)

Formula for the sum of the first natural numbers:

ⁿ∑_r=1 r = ½n(n + 1)

Formula for the sum of the squares of the first n natural numbers is:

ⁿ∑_r=1 r² = 1/6n(n + 1)(2n + 1)

Formula for the sum of the cubes of the first n natural numbers is:

ⁿ∑_r=1 r³ = ¼n²(n + 1)²

Rearrange ⁿ∑_r=1 k f(r) , with k being a constant:

k ⁿ∑_r=1 f(r)

Rearrange ⁿ∑_r=1 (f(r) + g(r)) :

ⁿ∑_r=1 f(r) + ⁿ∑_r=1 g(r)