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


Sequences and series are a core part of algebra, allowing us to solve problems involving patterns and summation. In this comprehensive guide, we will take a detailed look at the different types of sequences and series, how to identify their patterns, and the methods used to solve them. We will explore this topic with various examples and clear definitions to ensure a deeper understanding.

What are sequences?

A sequence is an ordered list of numbers that follows a particular pattern. Each number in the sequence is called a term. Sequences are often defined by a rule that tells how to find the next term from the previous terms. Let's take a deeper look at the types of sequences you may encounter.

Arithmetic sequence

An arithmetic sequence is a sequence of numbers in which the difference of any two successive members is a constant. This difference is called the "common difference" denoted by d. For example, the sequence:

2, 5, 8, 11, 14,...

is an arithmetic sequence whose common difference is d = 3.

The general term of an arithmetic sequence can be expressed using the following formula:

a n = a 1 + (n - 1) * d

where a n is the nth term, a 1 is the first term, and d is the common difference.

Visual example

2 5 8 11 14

Geometric progression

A geometric sequence is a sequence in which each term after the first term is found by multiplying the previous term by a fixed, non-zero number called the "common ratio," denoted by r. For example:

3, 6, 12, 24, 48,...

is a geometric sequence where the common ratio r = 2.

The general term of a geometric sequence can be written as:

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

where a n is the nth term, a 1 is the first term, and r is the common ratio.

Visual example

3 6 12 24 48

What are the series?

A series is the sum of the terms of a sequence. When you have a specific sequence and add up all of its terms, you create a series. Let's take a closer look at the types of series commonly studied in algebra.

Arithmetic series

An arithmetic series is the sum of the terms of an arithmetic sequence. The sum of the first n terms is denoted by S n and can be calculated using the formula:

S n = (n/2) * (a 1 + a n )

Alternatively, if you don't want to calculate a n, you can use:

S n = (n/2) * (2a 1 + (n - 1) * d)

where a 1 is the first term, a n is the nth term, and d is the common difference.

Example of arithmetic series

Consider the arithmetic sequence: 5, 8, 11, 14, 17 Find the sum of the first 5 terms.

a 1 = 5, d = 3, n = 5

S n = (5/2) * (2*5 + (5 - 1) * 3)
      = (5/2) * (10 + 12)
      = (5/2) * 22 = 55

Geometric series

The geometric series shows the sum of the terms in a geometric sequence. It can be expressed using the formula:

S n = A 1 * (1 - R n ) / (1 - R)

If r ≠ 1 and n is the number of terms. Where a 1 is the first term and r is the common ratio.

Example of geometric series

For the geometric sequence 3, 6, 12, 24..., find the sum of the first 4 terms.

a 1 = 3, r = 2, n = 4

S n = 3 * (1 - 2 4 ) / (1 - 2)
      = 3 * (1 - 16) / (-1)
      = 3 * (-15) / (-1)
      = 45

Special series

Let us consider some special types of chains that have unique properties and significance.

Finite series

A finite series has a fixed number of terms. Both arithmetic and geometric series can be finite.

Infinite series

Infinite series continue indefinitely. Geometric series are often used in the context of infinite series.

The formula for the sum of an infinite geometric series, where the absolute value of the common ratio |r| < 1, is given as follows:

S = A 1 / (1 - R)

Example of infinite series

Consider the series 1, 0.5, 0.25, 0.125,... whose common ratio r = 0.5.

S = 1 / (1 – 0.5)
  = 1 / 0.5
  = 2

Conclusion

Sequences and series are powerful mathematical concepts that allow us to recognize patterns and calculate sums quickly. Understanding arithmetic and geometric sequences and their related series provides a strong foundation for more advanced mathematical studies. By becoming familiar with the formulas and practicing with a variety of examples, solving problems involving sequences and series becomes a practical and manageable task.


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