4.2.8
Geometric Series
Geometric Series
Geometric Series
The sum of the terms in a geometric sequence is called a geometric series.
![Illustrative background for Series](https://image-v2.cdn.app.senecalearning.com/2020-01/734053bc-eb59-49fa-858e-9141962b24ca/geometric-series,h_400,q_80,w_640.png)
![Illustrative background for Series ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-01/734053bc-eb59-49fa-858e-9141962b24ca/geometric-series,h_400,q_80,w_640.png)
Series
Series
- The sum of the terms of a sequence is called a series.
- For the geometric sequence defined by , the first four terms are:
- The series is then given by:
- The sum of this series is 75.
![Illustrative background for General formula](https://image-v2.cdn.app.senecalearning.com/2020-01/39003e18-3a81-4732-95b4-85644f291c6e/geometric-series-2,h_400,q_80,w_640.png)
![Illustrative background for General formula ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-01/39003e18-3a81-4732-95b4-85644f291c6e/geometric-series-2,h_400,q_80,w_640.png)
General formula
General formula
- We can write the sum of the first terms of a geometric series with common ratio
as:
- Multiplying both sides of the equation by we have:
![Illustrative background for General formula](https://image-v2.cdn.app.senecalearning.com/2020-01/b710c44d-1eb7-478d-a590-91ca6cd7ddf2/geometric-series-3,h_400,q_80,w_640.png)
![Illustrative background for General formula ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-01/b710c44d-1eb7-478d-a590-91ca6cd7ddf2/geometric-series-3,h_400,q_80,w_640.png)
General formula
General formula
- Subtracting this from the original equation for , almost all of the terms cancel out:
- Taking out a factor of and dividing by
, we have the general equation for the sum of a geometric series:
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
![Go student ad image](/en-GB/revision-notes/_next/image?url=%2Fen-GB%2Frevision-notes%2Fimages%2Fgo-student-uk-ad.jpg&w=640&q=100)
Unlock your full potential with GoStudent tutoring
Affordable 1:1 tutoring from the comfort of your home
Tutors are matched to your specific learning needs
30+ school subjects covered