The sigma sign Σ is a short way of saying "add up every term made by the same rule, one after another".
Common sums such as the sum of the natural numbers from 1 to n, of their squares or of their cubes can be found at once with a single formula, without adding term by term.
Knowing where those formulas come from lets you break a complicated-looking sum into sums of k, k², k³ and constants.
In the first picture the heights of the bars add up to the value of Σ; in the second, two staircases of dots fit together into a rectangle, which shows why the sum of the natural numbers is n(n + 1)/2.
What the sigma sign means
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The heights of the n shaded bars add up to the Σ value at the top; the buttons below choose the term to add: k, k² or 2k − 1
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💡 Σ means "add them all up"
①k = 1 under the Σ is the starting index, n above it is the last index, and the expression on the right is the rule for each term
②So Σ of aₖ from k = 1 to n is a₁ + a₂ + … + aₙ
③In the picture the heights of the shaded bars add up to the Σ value at the top, and always match the formula value on the right
④Adding the first n odd numbers 1, 3, 5, … gives n²
The sum of natural numbers: two staircases of dots
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A staircase of 1, 2, …, n dots (gold) and the same staircase turned over (blue) form an n × (n + 1) rectangle
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Sum of Naturals
n(n+1)2
1 + 2 + 3 + … + n is half of a rectangle of n × (n + 1) dots
Three basic formulas
Σk
n(n+1)2
1 + 2 + … + n
Σk²
n(n+1)(2n+1)6
1² + 2² + … + n²
Σk³
[n(n+1)2]² = (Σk)²
1³ + 2³ + … + n³ = (1 + 2 + … + n)²
🔗 How the three formulas are connected
①The sum of cubes equals the square of the sum of the natural numbers; for example, 1³ + 2³ + 3³ = 36 = 6²
②With these three formulas and the constant sum n, most sums of polynomial terms can be found
③A complicated expression is first expanded and split into k³, k², k and constant terms
Properties of Σ
Linearity
Σ(cak + bk) = c·Σak + Σbk
Constant multiples come out of the Σ, and sums can be split up
Recovering the general term from the sum
an = Sn - Sn-1 (n ≥ 2)
Knowing the sum Sₙ, this gives the general term for n ≥ 2
⚠️ A caution when recovering the general term
①For n ≥ 2, aₙ = Sₙ − Sₙ₋₁
②For n = 1, use a₁ = S₁ separately
③Check whether the two results fit one formula; if not, write the first term separately
Put n = 10 into the formula for the sum of natural numbers, n(n + 1)/2.
Σk = 10 × 112
2
Divide 10 × 11 = 110 by 2.
= 1102 = 55
▸ 55
Pairing front and back, 1 with 10 and 2 with 9, gives five pairs that each add up to 11, so the sum is 55. The formula n(n + 1)/2 is this pairing written as an expression.
Example 2
Find the value of 1² + 2² + 3² + 4².
1
Put n = 4 into the formula for the sum of squares, n(n + 1)(2n + 1)/6.
Σk² = 4 × 5 × 96
2
Divide 4 × 5 × 9 = 180 by 6.
= 1806 = 30
▸ 30
Adding 1 + 4 + 9 + 16 directly also gives 30. The more terms there are, the more the formula saves.
2023 CSAT Math type, adapted
Write the sum of (2k + 1) from k = 1 to n as an expression in n.
①n(n+2)
②n(n+1)
③n²
④2n+1
⑤n²+1
▸ ① n(n+2)
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Splitting with the properties of Σ gives Σ(2k + 1) = 2Σk + Σ1, and Σ1 is 1 added n times, which is n.