Grade 11
  1. Algebra  1.1. Sequences and Series  1.1.1. Arithmetic Sequence  1.1.2. Arithmetic Series  1.1.3. Geometric Sequence  1.1.4. Geometric Series  1.1.5. Convergence and Divergence  1.1.6. Sum to Infinity  1.1.7. Sigma Notation  1.2. Complex Numbers  1.2.1. Imaginary and Complex Numbers  1.2.2. Operations with Complex Numbers  1.2.3. Polar and Cartesian Forms  1.2.4. Complex Conjugates  1.2.5. Modulus and Argument  1.2.6. De Moivre's Theorem  1.2.7. Powers and Roots of Complex Numbers  1.3. Binomial Theorem  1.3.1. Expansion of Binomials  1.3.2. General Term  1.3.3. Binomial Coefficients  1.3.4. Applications of Binomial Theorem  1.3.5. Pascal's Triangle
  2. Functions and Graphs  2.1. Types of Functions  2.1.1. Polynomial Functions  2.1.2. Rational Functions  2.1.3. Exponential Functions  2.1.4. Logarithmic Functions  2.1.5. Trigonometric Functions  2.1.6. Piecewise Functions  2.2. Transformations of Functions  2.2.1. Translations  2.2.2. Reflections  2.2.3. Dilations and Stretches  2.2.4. Rotations  2.3. Inverse Functions  2.3.1. Finding Inverses  2.3.2. Graphs of Inverse Functions  2.3.3. Properties of Inverse Functions  2.3.4. Restrictions for Inverses
  3. Trigonometry  3.1. Trigonometric Ratios and Identities  3.1.1. Basic Trigonometric Ratios  3.1.2. Pythagorean Identities  3.1.3. Sum and Difference Formulas  3.1.4. Double Angle Formulas  3.1.5. Half Angle Formulas  3.1.6. Product-to-Sum and Sum-to-Product Formulas  3.2. Trigonometric Equations  3.2.1. Solving Trigonometric Equations  3.2.2. General Solutions in Trigonometric Equations  3.3. Graphs of Trigonometric Functions  3.3.1. Sine Graph  3.3.2. Cosine Graph  3.3.3. Tangent Graph  3.3.4. Amplitude and Period Adjustments  3.3.5. Phase Shifts in Graphs of Trigonometric Functions  3.4. Applications of Trigonometry  3.4.1. Laws of Sines and Cosines  3.4.2. Area of Triangles  3.4.3. Solving Triangles  3.4.4. Bearings and Navigation
  4. Calculus  4.1. Limits and Continuity  4.1.1. Concept of a Limit  4.1.2. One-Sided Limits  4.1.3. Limits at Infinity  4.1.4. Continuity of a Function  4.1.5. Types of Discontinuities  4.2. Differentiation  4.2.1. Definition of Derivative  4.2.2. Rules of Differentiation  4.2.3. Higher Order Derivatives  4.2.4. Chain Rule  4.2.5. Product Rule  4.2.6. Quotient Rule  4.3. Applications of Differentiation  4.3.1. Rate of Change  4.3.2. Tangents and Normals  4.3.3. Maxima and Minima  4.3.4. Increasing and Decreasing Functions  4.3.5. Optimization Problems  4.3.6. Curve Sketching  4.3.7. Related Rates  4.4. Integration  4.4.1. Antiderivatives  4.4.2. Indefinite Integrals  4.4.3. Definite Integrals  4.4.4. Fundamental Theorem of Calculus  4.4.5. Techniques of Integration  4.4.6. Area Under a Curve  4.5. Applications of Integration  4.5.1. Area Between Curves  4.5.2. Volumes of Solids of Revolution  4.5.3. Average Value of a Function
  5. Vectors and Matrices  5.1. Vectors  5.1.1. Representation of Vectors  5.1.2. Vector Operations  5.1.3. Dot Product  5.1.4. Cross Product  5.1.5. Applications in Geometry  5.1.6. Projection of Vectors  5.2. Matrices  5.2.1. Types of Matrices  5.2.2. Matrix Operations  5.2.3. Determinants  5.2.4. Inverse of a Matrix  5.2.5. Solving Systems of Linear Equations  5.2.6. Cramer's Rule
  6. Probability and Statistics  6.1. Probability  6.1.1. Basic Probability Concepts  6.1.2. Conditional Probability  6.1.3. Independent and Dependent Events  6.1.4. Bayes' Theorem  6.1.5. Combinatorial Probability  6.2. Random Variables  6.2.1. Discrete Random Variables  6.2.2. Continuous Random Variables  6.2.3. Probability Distributions  6.3. Probability Distributions  6.3.1. Binomial Distribution  6.3.2. Normal Distribution  6.3.3. Poisson Distribution  6.3.4. Uniform Distribution  6.4. Statistics  6.4.1. Measures of Central Tendency  6.4.2. Measures of Dispersion  6.4.3. Data Collection and Representation  6.4.4. Correlation and Regression  6.4.5. Sampling Techniques
  7. Coordinate Geometry  7.1. Straight Lines  7.1.1. Slope and Intercept Form  7.1.2. Distance Formula  7.1.3. Midpoint Formula  7.1.4. Angle Between Lines  7.1.5. Equation of Lines  7.2. Conic Sections  7.2.1. Circle  7.2.2. Parabola  7.2.3. Ellipse  7.2.4. Hyperbola  7.2.5. Properties of Conics  7.2.6. Parametric Equations of Conics  7.2.7. Polar Coordinates of Conics
  8. Mathematical Reasoning  8.1. Logic  8.1.1. Statements and Logical Connectives  8.1.2. Truth Tables  8.1.3. Logical Equivalence  8.1.4. Quantifiers  8.1.5. Proof Techniques  8.2. Proofs  8.2.1. Direct Proof  8.2.2. Indirect Proof  8.2.3. Proof by Contradiction  8.2.4. Proof by Mathematical Induction

Grade 11 → Algebra → Sequences and Series ↓


Geometric Sequence


A geometric sequence, also known as a geometric progression, is a sequence of numbers where each term after the first term is found by multiplying the previous term by a constant, known as the common ratio. This concept is widely used in various mathematical analyses and has real-world applications in fields such as finance, computer science, and physics.

Basics of geometric sequence

The general form of a geometric sequence can be expressed as:

a, ar, ar^2, ar^3, ar^4, ..., ar^n

Here:

  • a is the first term of the sequence.
  • r is the common ratio, which is a constant.
  • n is the term number.

Each term is obtained by multiplying the previous term by r.

Example of a geometric sequence

Let us consider an example:

2, 6, 18, 54, ...

In this sequence, the first term is a 2 and the common ratio r is 3 To get the next term you multiply the previous term by 3.

Finding the common ratio

The common ratio can be determined by dividing any term in the sequence by its previous term. If the sequence is geometric, the common ratio must be the same for all successive pairs of terms. Formally,

r = a /a r = a /a

For example, in the sequence 2, 6, 18, 54 :

  • r = 6/2 = 3
  • r = 18/6 = 3
  • r = 54/18 = 3

General term of a geometric sequence

The n-th term of a geometric sequence can be found using the following formula:

a_n = a * r^(n-1)

Where:

  • a_n is the nth term.
  • a is the first term.
  • r is the common ratio.
  • n is the number of terms.

Working example

Suppose a geometric sequence begins with 5, and the common ratio is 2. To find the 5th term (a 5 ):

a_5 = 5 * 2^(5-1) = 5 * 16 = 80

Hence the 5th term is 80.

Sum of geometric sequence

The sum of the first n terms of a geometric sequence can be calculated using the following:

S_n = a * (1-r^n)/(1-r)

If the common ratio r is between -1 and 1 (except 1), or when the absolute value of r is less than 1, the infinite sum can be calculated using:

S = a / (1-r)

Example of a finite sum

Using the previous sequence 2, 6, 18, 54 , find the sum of the first 4 terms.

S_4 = 2 * (1-3^4)/(1-3) = 2 * (1-81)/(-2) = 2 * (80/2) = 80

Example of infinite sum

Consider an infinite geometric sequence 5, 2.5, 1.25, ... in which r = 0.5 :

S = 5 / (1-0.5) = 5 / 0.5 = 10

Graphical representation

Geometric sequences can be represented visually by plotting the terms on a graph, where the x-axis represents the term number and the y-axis represents the value of the term. A simple illustration is given below:

Term Number Duration value

Challenges and considerations

When working with geometric sequences, it is important to consider the following:

  • The common ratio r must not be zero, as this would make all subsequent terms zero.
  • If r is negative, the sequence will alternate between positive and negative values.

Example with a negative common ratio

Consider the sequence 4, -8, 16, -32, ... with a = 4 , r = -2 . The first few terms show an alternating pattern:

  • a_1 = 4
  • a_2 = 4 * (-2) = -8
  • a_3 = -8 * (-2) = 16
  • a_4 = 16 * (-2) = -32

Applications of geometric sequences

Geometric sequences are not just theoretical concepts, but also have important applications in the real world:

  • Finance: Used to model compound interest and investments.
  • Computer science: Algorithms that involve exponential growth or decrease.
  • Physics: Modeling of exponential decay processes.

Example in finance: compound interest

If you invest $1,000 at an annual interest rate of 5%, compounded annually, the sequence showing the balance each year is geometric:

1000, 1050, 1102.5, ...

Here, a = 1000 and r = 1.05.

Let us calculate the balance after 5 years:

a_5 = 1000 * 1.05^4 ≈ 1215.51

The remaining amount after 5 years would be approximately $1215.51.

Exploring variations in geometric sequences

Geometric sequences can be altered to represent more complex phenomena. Consider sequences with variable growth rates or target sequences that show specific growth patterns. By adjusting common ratios or initial positions, fascinating new patterns emerge, further expanding their applicability in modeling complex systems.

Conclusion

Understanding geometric sequences is an essential part of algebra that connects to many other mathematical fields and real-world applications. With a firm grasp on geometric sequences, one can tackle problems ranging from simple pattern recognition to complex financial modeling and beyond.


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Geometric Sequence

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