Precalculus — Chapter 12

Sigma Notation

Reading Σ notation, the 5 essential summation formulas, summation properties, writing series in sigma form — with worked examples.

What is Sigma Notation?

Σᵢ₌₁ⁿ aᵢ

= a₁ + a₂ + a₃ + … + aₙ

Σ (sigma)

Means "sum of" — capital Greek letter S for Sum

Index variable (i)

Counter that increases by 1 each step. Often i, k, or j

Lower bound (i = 1)

Starting value of the index — where counting begins

Upper bound (n)

Ending value of the index — where counting stops

Formula (aᵢ)

The expression evaluated at each index value

Number of terms

Upper bound − lower bound + 1 (e.g., i=1 to 5 gives 5 terms)

Reading Sigma Notation

Read Σᵢ₌₁ⁿ aᵢ as: “the sum of aᵢ, as i goes from 1 to n.”

Σᵢ₌₁⁴ i→1 + 2 + 3 + 4 = 10

Add each index value from 1 to 4

Σᵢ₌₁³ i²→1² + 2² + 3² = 1 + 4 + 9 = 14

Square each index value and sum

Σᵢ₌₂⁵ (3i − 1)→(3·2−1) + (3·3−1) + (3·4−1) + (3·5−1) = 5 + 8 + 11 + 14 = 38

Index starts at 2 (not 1) — still substitute each value

Σᵢ₌₁⁶ 7→7 + 7 + 7 + 7 + 7 + 7 = 42

Constant formula — result is always 7·6 = 42

The 5 Essential Summation Formulas

Memorize these. They let you evaluate sums of 100 or 1000 terms instantly.

Σᵢ₌₁ⁿ 1= n(Count of terms)

Adding 1 exactly n times gives n

Σᵢ₌₁¹⁰⁰ 1 = 100

Σᵢ₌₁ⁿ c= cn(Constant sum)

Any constant c, added n times, equals cn

Σᵢ₌₁⁸ 5 = 5·8 = 40

Σᵢ₌₁ⁿ i= n(n+1)/2(Sum of first n integers)

Gauss formula — the classic triangular number result

Σᵢ₌₁¹⁰⁰ i = 100·101/2 = 5050

Σᵢ₌₁ⁿ i²= n(n+1)(2n+1)/6(Sum of first n squares)

Quadratic formula — three-factor product divided by 6

Σᵢ₌₁⁸ i² = 8·9·17/6 = 204

Σᵢ₌₁ⁿ i³= [n(n+1)/2]²(Sum of first n cubes)

Square the triangular number formula — elegant result

Σᵢ₌₁⁵ i³ = [5·6/2]² = 15² = 225

Summation Rules (Properties)

These rules let you break apart or simplify complex sigma expressions before evaluating.

Scalar Multiplication:Σ c·aᵢ = c · Σ aᵢ

A constant multiplied inside the sum can be factored outside.

Σᵢ₌₁⁵ 3i = 3 · Σᵢ₌₁⁵ i = 3 · 15 = 45

Sum Rule:Σ(aᵢ + bᵢ) = Σ aᵢ + Σ bᵢ

A sum of two formulas splits into two separate sigma expressions.

Σᵢ₌₁³ (i + i²) = Σᵢ₌₁³ i + Σᵢ₌₁³ i² = 6 + 14 = 20

Difference Rule:Σ(aᵢ − bᵢ) = Σ aᵢ − Σ bᵢ

A difference of two formulas splits into two separate sigma expressions.

Σᵢ₌₁⁴ (i² − i) = Σᵢ₌₁⁴ i² − Σᵢ₌₁⁴ i = 30 − 10 = 20

Split Sum:Σᵢ₌₁ⁿ aᵢ = Σᵢ₌₁ᵏ aᵢ + Σᵢ₌ₖ₊₁ⁿ aᵢ

A sum can be split at any index k into two adjacent pieces.

Σᵢ₌₁¹⁰ i = Σᵢ₌₁⁵ i + Σᵢ₌₆¹⁰ i = 15 + 40 = 55

Worked Examples

Example 1: Evaluate Σᵢ₌₁⁵ (2i + 3)

Split using sum rule and scalar rule:

= 2 · Σᵢ₌₁⁵ i + Σᵢ₌₁⁵ 3

= 2 · [5·6/2] + 3·5

= 2 · 15 + 15

= 30 + 15

= 45

Verify by expanding: 5 + 7 + 9 + 11 + 13 = 45 ✓

Example 2: Evaluate Σᵢ₌₁¹⁰⁰ i using the formula

Use formula: Σᵢ₌₁ⁿ i = n(n+1)/2, with n = 100

= 100 · 101 / 2

= 10100 / 2

= 5050

Gauss reportedly computed this at age 10 — adding all terms by hand.

Example 3: Evaluate Σᵢ₌₁⁸ i² using the formula

Use formula: Σᵢ₌₁ⁿ i² = n(n+1)(2n+1)/6, with n = 8

= 8 · 9 · (2·8 + 1) / 6

= 8 · 9 · 17 / 6

= 1224 / 6

= 204

Check: 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 = 204 ✓

Example 4: Write 1 + 4 + 9 + 16 + 25 in sigma notation

Identify the pattern: 1 = 1², 4 = 2², 9 = 3², 16 = 4², 25 = 5²

Each term is i² where i goes from 1 to 5 — five terms total.

Σᵢ₌₁⁵ i²

Example 5: Write the arithmetic series 3 + 7 + 11 + … + 43 in sigma notation

Find the pattern: first term a₁ = 3, common difference d = 4

General term: aᵢ = 3 + (i − 1)·4 = 4i − 1

Find n: 4n − 1 = 43 → 4n = 44 → n = 11

So i runs from 1 to 11.

Σᵢ₌₁¹¹ (4i − 1)

Check endpoints: i=1 → 4(1)−1 = 3 ✓ i=11 → 4(11)−1 = 43 ✓

Converting Between Forms

Sigma → Expanded Form

Identify the index, lower bound, and upper bound
Substitute each integer value of the index into the formula
Write out each resulting term separated by + (or −)
Optionally evaluate the full sum

Σᵢ₌₂⁴ (i² − 1)
= (4−1) + (9−1) + (16−1)
= 3 + 8 + 15 = 26

Expanded Form → Sigma

Find the pattern relating each term to its position
Write a formula for the nth term using index i
Count the terms to find the upper bound
Write Σᵢ₌₁ⁿ (formula)

2 + 5 + 8 + 11 + 14
aᵢ = 3i − 1, i = 1 to 5
→ Σᵢ₌₁⁵ (3i − 1)

Sigma Notation & Arithmetic / Geometric Series

Arithmetic Series

Σᵢ₌₁ⁿ [a₁ + (i−1)d] = n/2 · (a₁ + aₙ)

The sigma expression matches the arithmetic series formula Sₙ = n/2·(first + last).

Geometric Series

Σᵢ₌₁ⁿ a·rⁱ⁻¹ = a · (1 − rⁿ) / (1 − r), r ≠ 1

Each term is a·rⁱ⁻¹. The sigma index controls the exponent, and the closed-form formula evaluates the whole sum at once.

Key insight: Sigma notation is just a compact way of writing any series — arithmetic, geometric, or other. The summation formulas turn those compact expressions into single values without expanding every term.

Frequently Asked Questions

What does sigma (Σ) mean in math?

Sigma (Σ) is the Greek capital letter used to denote summation — it means 'add up a list of values.' The notation Σᵢ₌₁ⁿ aᵢ means a₁ + a₂ + a₃ + ... + aₙ. The variable below Σ is the index (often i, k, or j), the number below is the starting value, and the number above is the ending value. The formula to the right of Σ describes each term.

How do you evaluate sigma notation?

To evaluate sigma notation, substitute each integer value of the index into the formula and add the results. For example, Σᵢ₌₁⁵ (2i + 3) means plug in i = 1, 2, 3, 4, 5 and sum: (2·1+3) + (2·2+3) + (2·3+3) + (2·4+3) + (2·5+3) = 5 + 7 + 9 + 11 + 13 = 45. For large sums, use the standard summation formulas to avoid expanding every term.

What are the essential summation formulas to memorize?

The five essential summation formulas are: (1) Σ 1 = n (count of terms), (2) Σ i = n(n+1)/2 (sum of first n integers), (3) Σ i² = n(n+1)(2n+1)/6 (sum of first n squares), (4) Σ i³ = [n(n+1)/2]² (sum of first n cubes), (5) Σ c = cn (sum of a constant). These all assume the index runs from i = 1 to n.

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