4.2.10
Modelling with Sequences & Series
Modelling with Sequences and Series
Modelling with Sequences and Series
We can use sequences and series to model numerical problems that occur in real life.
Fixed amount
Fixed amount
- Numbers that increase by fixed amounts can be modelled by arithmetic sequences.
Example
Example
- On the Sunday after a minor surgery, a woman is able to walk a half-mile. Each Sunday, she walks an additional quarter-mile.
- After 8 weeks, what will be the total number of miles she has walked?
Arithmetic series
Arithmetic series
- The number of miles the woman walks increases by a fixed amount each week.
- This means we can model this as an arithmetic series with and .
- Using the equation for the sum of an arithmetic series, we have:
- So the woman has walked a total of 11 miles after her surgery.
Fixed percentage
Fixed percentage
- Numbers that increase by a fixed percentage or fraction can be modelled by geometric sequences.
Example
Example
- A pendulum travels a distance of 48 cm on its first swing. On each successive swing, it travels of the distance of the previous swing.
- What is the total distance travelled by the pendulum when it stops swinging?
Geometric series
Geometric series
- The distance of the swing decreases by a fixed fraction.
- That means we can model total distance of the pendulum as a geometric series with and .
- Since , we know that the infinite series is convergent and that the pendulum's distance will diminish until it eventually stops.
- Finding the sum of the infinite series will give the total distance travelled by the pendulum.
Sum of infinite series
Sum of infinite series
- Using the equation for the sum of an infinite geometric series, we have:
- So the pendulum travels a total distance of 192 cm before it stops.
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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