Basic Mathematics
1 Arithmetic as Structure
1. 1.1 Why Arithmetic Has Rules
2. 1.2 Sets: The Language of Structure
3. 1.3 Properties of Addition
4. 1.4 Properties of Multiplication
5. 1.5 Negation and Additive Inverses
6. 1.6 Subtraction and Integer Arithmetic
7. 1.7 Reciprocals and Multiplicative Inverses
8. 1.8 Division and Fractions
9. 1.9 Exponents: Repeated Multiplication
10. 1.10 Zero and Negative Exponents
11. 1.11 Exploring Structure: Symmetry, Clocks, and Composition
12. 1.12 Order of Operations and Structure Summary
2 Algebraic Foundations
1. 2.1 What is a Variable?
2. 2.2 Combining Like Terms
3. 2.3 Distribution and Factoring
4. 2.4 What is an Equation?
5. 2.5 Solving Multi-Step Linear Equations
6. 2.6 Special Cases: No Solution and Infinite Solutions
7. 2.7 Word Problems I: Translating to Equations
8. 2.8 Word Problems II: Geometry and Rate Problems
9. 2.9 Inequalities
10. 2.10 Polynomials and Like Terms
11. 2.11 Polynomial Multiplication and Factoring I
12. 2.12 Factoring II and Part II Summary
3 Number Theory and Patterns
1. 3.1 Divisibility Rules and Tests
2. 3.2 Primes and Composite Numbers
3. 3.3 Prime Factorization and Fundamental Theorem
4. 3.4 GCD and LCM
5. 3.5 The Euclidean Algorithm
6. 3.6 Introduction to Modular Arithmetic
7. 3.7 Computing with Congruences
8. 3.8 Linear Congruences and Diophantine Equations
9. 3.9 Patterns in Number Theory
10. 3.10 Part III Summary: Number Theory as Structure
4 Geometric Reasoning
1. 4.1 Points, Lines, and Planes: The Undefined Terms
2. 4.2 Angles: Measuring and Classifying
3. 4.3 Parallel Lines and Transversals
4. 4.4 Triangles: Classification and Angle Sum
5. 4.5 Congruent Triangles
6. 4.6 Proof Techniques in Geometry
7. 4.7 Quadrilaterals and Their Properties
8. 4.8 Perimeter and Area: Rectangles, Triangles, Parallelograms
9. 4.9 The Pythagorean Theorem
10. 4.10 Similar Triangles
11. 4.11 Circles: Definitions and Basic Properties
12. 4.12 Part IV Summary: Geometry as Logical Structure
5 Counting, Probability and Combinatorics
1. 5.1 Counting Principles: Addition and Multiplication
2. 5.2 Permutations
3. 5.3 Combinations
4. 5.4 Pascal’s Triangle and Binomial Theorem
5. 5.5 Counting with Restrictions and Casework
6. 5.6 Introduction to Probability
7. 5.7 Compound Events: AND, OR, Conditional
8. 5.8 Expected Value
9. 5.9 Geometric Probability and Continuous Outcomes
10. 5.10 Part V Summary: Counting as Structure
6 Functions, Graphs and Synthesis
1. 6.1 What is a Function?
2. 6.2 Linear Functions and Their Graphs
3. 6.3 Systems of Linear Equations
4. 6.4 Quadratic Functions and Parabolas
5. 6.5 Introduction to Coordinate Geometry
6. 6.6 Transformations of Functions
7. 6.7 Function Composition and Inverses
8. 6.8 Course Summary: Mathematics as Structure
Integrated Maths I
7 Sets and Proof
1. 7.1 Sets: Notation, Subsets and Operations
2. 7.2 Intuitive Proofs: Dominoes on a Chessboard
8 Equations, Functions and Graphs
1. 8.1 Solving Equations and Inequalities
2. 8.2 Functions: Definition, Notation and Domain
3. 8.3 Properties of Functions from Their Graphs
4. 8.4 Piecewise Functions, Basic Graphs and Transformations
5. 8.5 Special Classes of Functions
6. 8.6 More Piecewise-Defined Functions
7. 8.7 Functional Equations
9 Quadratics
10 Polynomials
11 Sequences, Series and Induction
1. 11.1 Arithmetic Sequences
2. 11.2 Arithmetic Series
3. 11.3 Geometric Sequences
4. 11.4 Mathematical Induction
5. 11.5 More Sequences and Series
12 Exponents, Logarithms and Radicals
1. 12.1 Exponents and Logarithms
2. 12.2 Radicals
13 Identities and Inequalities
1. 13.1 Identities and Manipulations
2. 13.2 Inequalities
3. 13.3 More Inequalities
4. 13.4 Some Advanced Strategies
14 Graph Theory, Combinatorics and Probability
1. 14.1 Graph Theory
2. 14.2 Combinatorics
3. 14.3 Probability and Counting
Integrated Maths II
15 Trigonometry
16 Parametric Equations, Polar Coordinates and Complex Numbers
17 The Real Numbers
18 Groups, Rings and Fields
1. 18.1 Integral Domains: Definition and Elementary Properties
19 Vector Spaces and Topology
1. 19.1 Vector Spaces
2. 19.2 Topology
Integrated Maths III and IV
20 Integrated Maths III
1. 20.1 Advanced Linear Algebra
2. 20.2 Advanced Group Theory
3. 20.3 Real Analysis
4. 20.4 Topology
21 Integrated Maths IV
1. 21.1 Complex Analysis
2. 21.2 Differential Equations
3. 21.3 Lie Groups
4. 21.4 Geometry
5. 21.5 Differential Geometry

notes / Integrated Mathematics / chapter 11: Sequences, Series and Induction

11.3 Geometric Sequences

In progress: these notes stop partway.

Now each term is a fixed multiple of the one before it.

Definition 11.3.1. A geometric sequence (or geometric progression) is a sequence where each term, after the first, is (a ratio) times the previous term.

Problem 11.3.2. The first term of a geometric sequence is , with common ratio .

Show solution

term	value

Therefore is the th term, and

Problem 11.3.3. The first term of a geometric progression is and the 4th term is . Find the 10th term.

Show solution

term	value

Given that is the th term of a geometric progression: , , so the 10th term is .

Problem 11.3.4. Geometric mean. Given positive numbers , their geometric mean is . Show that if , then for the sequence the GM of the terms of the sequence is equal to the GM of the first term and the last term.

Show solution

So is the GM of the first and last term:

The terms and the GM of the first of them:

	term	GM of the first terms