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Sigma (Math)

See how to solve the sigma (∑) of a sequence
(high school Sequence chapter).
8 examples and their solutions.

Sigma: Definition

Definition

k = 1nak = a1 + a2 + a3 + ... + an
Sigma, ∑, is a way
to write the sum of a sequence (series).
k = 1n ak is read as
[the sum of ak as k goes from 1 to n].

Example

43 + 53 + 63 + ... + 193
→ Sigma notation
Solution

Example

k = 1991k(k + 1)
Solution

Sigma: Property

Property

k = 1nc = cn
k = 1ncak = ck = 1nak
k = 1n(ak ± bk) = k = 1nak ± k = 1nbk

Example

k = 1nak = 7, k = 1nbk = -3
k = 1n(5ak + 2bk) = ?
Solution

Example

k = 1nak = 5
k = 1n(2ak - 7) = ?
Solution

Example

k = 1nak = n2 + 1
k = 1nnak = ?
Solution

Sigma k

Formula

k = 1nk = n(n + 1)2

Example

k = 1n(4k - 3)
Solution

Sigma k2

Formula

k = 1nk2 = n(n + 1)(2n + 1)6

Example

k = 1nk(3k + 1)
Solution

Sigma k3

Formula

k = 1nk3 = (n(n + 1)2)2

Example

k = 1nk(k2 + n)
Solution