4.2.6
Sigma Notation
Sigma Notation
Sigma Notation
The Greek letter Sigma Σ is used to represent the sum of a collection of terms.
![Illustrative background for Sigma](https://image-v2.cdn.app.senecalearning.com/2020-06/374ad281-aff2-475c-a2a9-3147965f177b/Screenshot%202020-06-16%20at%2014.11.56-min,h_400,q_80,w_640.png)
![Illustrative background for Sigma ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-06/374ad281-aff2-475c-a2a9-3147965f177b/Screenshot%202020-06-16%20at%2014.11.56-min,h_400,q_80,w_640.png)
Sigma
Sigma
- Sigma notation, Σ, is a shorthand notation for sums in which each term can be written as a general expression.
- We write an integer variable below Σ and give it a starting value.
- The highest value of is written above the Σ.
- The general expression is written to the right of Σ which is usually a function of .
- The highest and lowest values of are called the limits of the sum.
![Illustrative background for Evaluating sums](https://image-v2.cdn.app.senecalearning.com/2020-06/bd024919-1da3-44a9-b57b-f3d882b5616e/Screenshot%202020-06-16%20at%2014.11.59-min,h_400,q_80,w_640.png)
![Illustrative background for Evaluating sums ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-06/bd024919-1da3-44a9-b57b-f3d882b5616e/Screenshot%202020-06-16%20at%2014.11.59-min,h_400,q_80,w_640.png)
Evaluating sums
Evaluating sums
- To evaluate the sum, we work out the first term by substituting the initial value of into the general expression.
- The next term is found by adding 1 to the initial value of , and then substituting it in to the general expression.
- The value of increases by 1 for each term, and the sum ends when is equal to the highest value.
![Illustrative background for Example](https://image-v2.cdn.app.senecalearning.com/2020-06/9df8f22b-cca7-4dbf-98f8-2711481d40de/Screenshot%202020-06-16%20at%2014.12.00-min,h_400,q_80,w_640.png)
![Illustrative background for Example ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-06/9df8f22b-cca7-4dbf-98f8-2711481d40de/Screenshot%202020-06-16%20at%2014.12.00-min,h_400,q_80,w_640.png)
Example
Example
- Evaluate the sum .
![Illustrative background for Limits](https://image-v2.cdn.app.senecalearning.com/2020-06/63c0e8da-5eba-45d0-a6f1-aea7e8068399/Screenshot%202020-06-16%20at%2014.12.02-min,h_400,q_80,w_640.png)
![Illustrative background for Limits ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-06/63c0e8da-5eba-45d0-a6f1-aea7e8068399/Screenshot%202020-06-16%20at%2014.12.02-min,h_400,q_80,w_640.png)
Limits
Limits
- The limits of the sum show that we have to add the terms up for 2, 3 and 4.
![Illustrative background for First term](https://image-v2.cdn.app.senecalearning.com/2020-06/d9ec5918-11a6-4686-b4a0-83e7c2e4f7ae/Screenshot%202020-06-16%20at%2014.12.05-min,h_400,q_80,w_640.png)
![Illustrative background for First term ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-06/d9ec5918-11a6-4686-b4a0-83e7c2e4f7ae/Screenshot%202020-06-16%20at%2014.12.05-min,h_400,q_80,w_640.png)
First term
First term
- Substituting into
gives the first term:
![Illustrative background for Second term](https://image-v2.cdn.app.senecalearning.com/2020-06/9bf92373-6a1b-4e13-9172-37f943403c1c/Screenshot%202020-06-16%20at%2014.12.06-min,h_400,q_80,w_640.png)
![Illustrative background for Second term ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-06/9bf92373-6a1b-4e13-9172-37f943403c1c/Screenshot%202020-06-16%20at%2014.12.06-min,h_400,q_80,w_640.png)
Second term
Second term
- Substituting into
gives the second term:
![Illustrative background for Final term](https://image-v2.cdn.app.senecalearning.com/2020-06/8bfbb2ae-a12a-44b1-8937-bf93077e0789/Screenshot%202020-06-16%20at%2014.12.08-min,h_400,q_80,w_640.png)
![Illustrative background for Final term ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-06/8bfbb2ae-a12a-44b1-8937-bf93077e0789/Screenshot%202020-06-16%20at%2014.12.08-min,h_400,q_80,w_640.png)
Final term
Final term
- Substituting into
gives the third term:
- This is the final term, as is the upper limit of the sum.
![Illustrative background for Find the sum](https://image-v2.cdn.app.senecalearning.com/2020-06/364a2512-55e9-4ad6-931e-a0c9c47ed601/Screenshot%202020-06-16%20at%2014.12.09-min,h_400,q_80,w_640.png)
![Illustrative background for Find the sum ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-06/364a2512-55e9-4ad6-931e-a0c9c47ed601/Screenshot%202020-06-16%20at%2014.12.09-min,h_400,q_80,w_640.png)
Find the sum
Find the sum
- Adding each term together gives the value of the sum:
![Illustrative background for Sum of one](https://image-v2.cdn.app.senecalearning.com/2020-06/fc107998-95b4-47b3-91de-fc474aca9850/Screenshot%202020-06-16%20at%2014.18.07-min,h_400,q_80,w_640.png)
![Illustrative background for Sum of one ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-06/fc107998-95b4-47b3-91de-fc474aca9850/Screenshot%202020-06-16%20at%2014.18.07-min,h_400,q_80,w_640.png)
Sum of one
Sum of one
- An important sum you need to remember is .
- We can rewrite this as where there are terms, each equal to 1.
- This means the sum is adding the number 1 to itself times, which is equal to .
1Proof
1.1Types of Numbers
1.2Notation
2Algebra & Functions
2.1Powers & Roots
2.2Quadratic Equations
2.3Inequalities
2.4Polynomials
2.5Graphs
2.7Transformation of Graphs
3Coordinate Geometry
3.1Straight Lines
3.2Circles
3.2.1Equations of Circles centred at Origin
3.2.2Finding the Centre & Radius
3.2.3Equation of a Tangent
3.2.4Circle Theorems - Perpendicular Bisector
3.2.5Circle Theorems - Angle at the Centre
3.2.6Circle Theorems - Angle at a Semi-Circle
3.2.7Equation of a Perpendicular Bisector
3.2.8Equation of a Circumcircle
3.2.9Circumcircle of a Right-angled Triangle
3.3Parametric Equations (A2 only)
4Sequences & Series
4.1Binomial Expansion
5Trigonometry
5.2Trigonometric Functions
5.3Triangle Rules
6Exponentials & Logarithms
6.1Exponentials & Logarithms
7Differentiation
7.1Derivatives
7.2Graphs & Differentiation
7.3Differentiation With Trigonometry and Exponentials
7.4Rules of Differetiation (A2 only)
7.5Parametric & Implicit Differentiation
8Integration
8.1Integration
9Numerical Methods
9.1Finding Solutions
9.2Finding the Area
10Vectors
10.12D Vectors
10.23D Vectors
10.3Vector Proofs
Jump to other topics
1Proof
1.1Types of Numbers
1.2Notation
2Algebra & Functions
2.1Powers & Roots
2.2Quadratic Equations
2.3Inequalities
2.4Polynomials
2.5Graphs
2.7Transformation of Graphs
3Coordinate Geometry
3.1Straight Lines
3.2Circles
3.2.1Equations of Circles centred at Origin
3.2.2Finding the Centre & Radius
3.2.3Equation of a Tangent
3.2.4Circle Theorems - Perpendicular Bisector
3.2.5Circle Theorems - Angle at the Centre
3.2.6Circle Theorems - Angle at a Semi-Circle
3.2.7Equation of a Perpendicular Bisector
3.2.8Equation of a Circumcircle
3.2.9Circumcircle of a Right-angled Triangle
3.3Parametric Equations (A2 only)
4Sequences & Series
4.1Binomial Expansion
5Trigonometry
5.2Trigonometric Functions
5.3Triangle Rules
6Exponentials & Logarithms
6.1Exponentials & Logarithms
7Differentiation
7.1Derivatives
7.2Graphs & Differentiation
7.3Differentiation With Trigonometry and Exponentials
7.4Rules of Differetiation (A2 only)
7.5Parametric & Implicit Differentiation
8Integration
8.1Integration
9Numerical Methods
9.1Finding Solutions
9.2Finding the Area
10Vectors
10.12D Vectors
10.23D Vectors
10.3Vector Proofs
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