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

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

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1

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

ⁿ∑r=1 1 = n

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2

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-1r=1 f(r)

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3

Formula for the sum of the first natural numbers:

ⁿ∑r=1 r = ½n(n + 1)

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4

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

ⁿ∑r=1 r² = 1/6n(n + 1)(2n + 1)

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5

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

ⁿ∑r=1 r³ = ¼n²(n + 1)²

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6

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

k ⁿ∑r=1 f(r)

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7

Rearrange ⁿ∑r=1 (f(r) + g(r)) :

ⁿ∑r=1 f(r) + ⁿ∑r=1 g(r)

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