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 Series


In mathematics, a series is the sum of the terms of a sequence. A geometric series is a type of series where each term is a constant multiple of the previous term. This constant is known as the "common ratio." Through this article, we will explore what a geometric series is, and how we can find the sum of such series. We will also look at examples to help strengthen understanding.

Definition of geometric series

A geometric progression is one in which the ratio between successive terms remains constant. The general form of a geometric progression can be expressed as:

a, ar, ar 2 , ar 3 , ..., ar n-1

Here, a is the first term of the series, r is the common ratio, and n is the number of terms.

Sum of geometric series

The sum of the first n terms of a geometric progression is given by the formula:

S n = a(1 - r n ) / (1 - r)

The following conditions have to be kept in mind:

  • If |r| < 1 , then the series converges and the above formula applies.
  • If |r| > 1 , then the series does not converge.

Let's derive the formula step by step. Consider the series:

S = a + ar + ar 2 + ... + ar n-1

Multiply the entire series by r :

rS = ar + ar 2 + ar 3 + ... + ar n

Subtract the second equation from the first equation:

S - rS = a - ar n

Factor out S and solve:

S(1 - r) = a(1 - r n )

Thus, dividing both sides by (1 - r) , we get:

S = a(1 - r n ) / (1 - r)

Example

Example 1: Finite geometric series

Find the sum of the geometric series: 3, 6, 12, 24 .

In this instance:

  • First term a = 3
  • Common ratio r = 2 (since 6 / 3 = 2 )
  • Number of items n = 4

Applying the formula for summation:

S n = 3(1 - 2 4 ) / (1 - 2)

S n = 3(1 - 16)/(-1)

S n = 3(-15)/(-1)

S n = 45

Thus, the sum of the series is 45 .

Example 2: Infinite geometric series

Discuss the series: 8, 4, 2, 1, ... continuing to infinity.

In this example, the common ratio r = 0.5 (since 4 / 8 = 0.5 ). This is a geometric sequence where |r| < 1 . The series will converge and we can find its infinite sum.

The formula for the sum of an infinite geometric series is:

S = a / (1 - r)

Thus, for our series:

S = 8 / (1 - 0.5)

S = 8 / 0.5

S = 16

The sum of the infinite series is 16 .

Visualization of geometric series

Let us next imagine a geometric series. Consider the finite series 1 + 2 + 4 + 8 Here a = 1 and r = 2 , where n = 4 .

1 2 4 8 | | | | *------------*------------*------------*--------------->
1 2 4 8 | | | | *------------*------------*------------*--------------->

The points above show how each term is twice the previous term. The sum of this series is:

S 4 = 1(1 - 2 4 ) / (1 - 2)

On calculation, S 4 = 1(-15)/(-1) = 15 .

Real-life applications

Geometric series appear in a variety of real-life situations, such as finance for calculating compound interest, physics for signal processing, and computer science for analyzing algorithms.

Example: Compound interest

When money is invested at compound interest, it earns interest on both the principal amount and the interest accumulated in previous periods. The total amount after n years can be modeled using a geometric series.

If P is the principal amount, r is the interest rate per period, and n is the number of periods:

A = P(1 + r) n

Thus, compound interest itself forms a geometric series.

Consider an example where $100 is invested at an interest rate of 5% per annum for 3 years. Here:

  • P = 100
  • r = 0.05
  • n = 3

The amount after 3 years will be:

A = 100(1 + 0.05) 3

= 100(1.157625) = 115.76

The investment grew to $115.76 .

Summary

To summarize, we have covered the essential aspects of geometric series, from their definition and mathematical formulation to real-world applications and visualization. With a fundamental knowledge of geometric series, students can solve various mathematical problems with confidence. It is important to understand both finite and infinite geometric series, as they provide valuable insights into patterns of growth and decay in many topics.

Engaging with more complex problems will further enhance learning, and help us understand the broader role of geometric categories in a number of contexts.


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Geometric Series

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