Sigma Notation

Sigma notation is a way to represent the summation of a series of numbers. It uses the Greek letter Σ (sigma) to indicate a sum.

General Form

The general form of sigma notation is:

Σi=mn f(i)

Where:

Interactive Example







Click for Help: Understanding Sigma Notation

1. Enter a lower limit and upper limit to define the range of summation.

2. Provide a function in terms of i (e.g., i^2 for the square of i).

3. Click Calculate Sum to see the result.

Sigma Notation

Sigma notation is a mathematical way to represent the summation of a series of terms. It uses the Greek letter Σ to denote the sum.

General Form

Σi=mn f(i)

Where:

Examples

Summation of Integers

Calculate Σi=15 i:

1 + 2 + 3 + 4 + 5 = 15

Summation of Squares

Calculate Σi=14 i2:

12 + 22 + 32 + 42 = 1 + 4 + 9 + 16 = 30

Properties of Summation

Applications of Sigma Notation

Sigma Notation

Properties of Summation

Summation Calculator

Enter the constants and limits to calculate the summation:

Result: 0

Quiz on Properties of Summation

What is the result of Σi=13(2i + 1)?

Practical Applications of Sigma Notation

Sigma Notation Calculator

Calculate the summation of a function:





Result: 0

Sigma Notation (Σ)

Sigma notation is a compact mathematical way to represent the sum of many terms. Instead of writing long additions, we use the Greek letter Σ.

General Form of Sigma Notation

Σi=mn f(i)

This means: start at i = m, substitute values into f(i), and add all results until i = n.

Basic Examples

Example 1: Sum of Natural Numbers

Evaluate:

Σi=15 i

Solution:

1 + 2 + 3 + 4 + 5 = 15

Example 2: Sum of Squares

Σi=14 i²
1² + 2² + 3² + 4² = 1 + 4 + 9 + 16 = 30

Example 3: Constant Terms

Σi=16 3

The number 3 appears 6 times:

3 + 3 + 3 + 3 + 3 + 3 = 18

Algebraic Sigma Expressions

Example 4: Linear Expression

Σi=14 (2i + 1)

Expand each term:

(2·1 + 1) + (2·2 + 1) + (2·3 + 1) + (2·4 + 1)
= 3 + 5 + 7 + 9 = 24

Example 5: Negative & Mixed Terms

Σi=13 (5 − i)
(5−1) + (5−2) + (5−3) = 4 + 3 + 2 = 9

Important Sigma Formulas (Exam Ready)

Sum of first n natural numbers

Σi=1n i = n(n + 1) / 2

Sum of squares

Σi=1n i² = n(n + 1)(2n + 1) / 6

Sum of cubes

Σi=1n i³ = [n(n + 1)/2]²

Properties of Sigma Notation

Interactive Sigma Calculator







Applications of Sigma Notation

Sigma notation is essential for higher mathematics, especially calculus, sequences, series, and computer science.

Changing the Index of Summation

Sometimes sigma notation does not start at 1. You must be careful when the lower limit changes.

Example 6: Index Starting at 0

Σi=04 (2i)

Solution:

2(0) + 2(1) + 2(2) + 2(3) + 2(4)
= 0 + 2 + 4 + 6 + 8 = 20

⚠️ Always substitute the correct starting value of i.

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Combining Sigma Expressions

Example 7: Splitting a Sum

Σi=15 (i + 3)

Step 1: Split the sum

Σ i + Σ 3

Step 2: Evaluate each part

Σi=15 i = 15
Σi=15 3 = 3 × 5 = 15

Final Answer:

15 + 15 = 30
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Repeated Multiplication (Geometric-Type Sums)

Example 8: Powers of 2

Σi=04 2i
2⁰ + 2¹ + 2² + 2³ + 2⁴
= 1 + 2 + 4 + 8 + 16 = 31

These types of sums often appear in computer science and sequences.

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Common Exam Mistakes to Avoid

Tip: Always write out at least the first two terms in exams.
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Exam-Style Practice Questions

Question 1

Evaluate: Σi=16 (i − 1)

Show Solution
(1−1) + (2−1) + (3−1) + (4−1) + (5−1) + (6−1)
= 0 + 1 + 2 + 3 + 4 + 5 = 15

Question 2

Evaluate: Σi=14 (3i)

Show Solution
3(1) + 3(2) + 3(3) + 3(4)
= 3 + 6 + 9 + 12 = 30
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Real-Life Interpretation

Sigma notation can represent real-world totals.

Example 9: Saving Money

You save R10 more each week than the previous week.

Week 1: R10
Week 2: R20
Week 3: R30
Week 4: R40

This can be written as:

Σi=14 10i

Total Saved:

10(1 + 2 + 3 + 4) = 10 × 10 = R100
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Summary

Mastering sigma notation makes later topics like sequences and series much easier.