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 ↓


Sigma Notation


Sigma notation is a concise and powerful way to represent the sum of multiple terms. It is widely used in mathematics to work with sequences and series. In grade 11 math, understanding sigma notation is important because it simplifies the process of working with long expressions involving series, especially when calculating the sum of a large number of terms.

Understanding the basics of sigma notation

Sigma notation uses the Greek letter Σ (sigma) to represent summation. The general form of sigma notation is:

Σ (from i=a to b) of f(i)

This expression is read as "the sum of f(i) from i equals a to i equals b". Here, i is the index of the sum, a is the lower limit, and b is the upper limit. The function f(i) represents the terms that are to be added together. Let us understand this in more detail.

Components of sigma notation

  • Sum index (i): The variable i is an index that starts from the lower limit a and increases by 1 until it reaches the upper limit b.
  • Lower limit (a): The value at which the index i starts.
  • Upper limit (b): The value at which the index i stops.
  • Function f(i): Expression defined in terms of the summation index. Each value of i is substituted into this function to generate each term of the series.

Visual example of sigma notation

Let's see how sigma notation works by using a simple addition:

Σ (from i=1 to 4) of i

This notation represents the sum of all natural numbers from 1 to 4. Divide it:

Σ i=1 4 I , 1 + 2 + 3 + 4 = 10

Reading from left to right: set i to 1, calculate the term (which is simply i), then increase i to 2, and repeat until i is 4. Next, sum all of these terms.

Example 1:

Consider the series of even numbers from 2 to 10. Using sigma notation it is represented as:

Σ (from i=1 to 5) of (2i)

Expanding this gives:

2(1) + 2(2) + 2(3) + 2(4) + 2(5) = 2 + 4 + 6 + 8 + 10 = 30

Properties of sigma notation

  • Linearity of addition: If a and b are constants, then: 
    Σ (from i=a to b) of [c * f(i) + d * g(i)] = c * Σ (from i=a to b) of f(i) + d * Σ (from i=a to b) of g(i)
  • Division of Amount: The amount may be divided into the following parts: 
    Σ (from i=a to b) of f(i) = Σ (from i=a to c) of f(i) + Σ (from i=c+1 to b) of f(i)
  • Combining totals: If the indices and limits match, totals can be combined: 
    Σ (from i=a to b) of f(i) + Σ (from i=a to b) of g(i) = Σ (from i=a to b) of [f(i) + g(i)]

Working with arithmetic and geometric series

Arithmetic series in sigma notation

An arithmetic series is a sequence of numbers in which each term increases by a constant value. The nth term of an arithmetic sequence can be expressed as:

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

where a is the first term, and d is the common difference between the terms. In sigma notation, an arithmetic series is represented as follows:

Σ (i=1 to n) [a + (i-1)d]

Example 2:

Consider the arithmetic series: 3, 6, 9, 12, ... up to 15 terms.

The first term is a 3 and the common difference is d 3.

Express this in sigma notation:

Σ (from i=1 to 15) [3 + (i-1)*3]

This gives us the sum of the arithmetic series.

Geometric series in sigma notation

A geometric series is a sequence of numbers where each term is multiplied by a constant factor. The nth term of a geometric sequence can be expressed as:

a(n) = ar^(n-1)

where a is the first term, and r is the common ratio. In sigma notation, the geometric series is represented as:

Σ (from i=1 to n) [ar^(i-1)]

Example 3:

Consider the geometric series: 5, 10, 20, 40, up to 4 terms.

The first term is a 5 and the common ratio is r 2.

Express this in sigma notation:

Σ (from i=1 to 4) [5 * 2^(i-1)]

This gives us the sum of the geometric series.

Examples and applications

Example 4:

Find the sum of the first 10 odd numbers using sigma notation.

Σ (from i=1 to 10) [2i - 1]

Expanding this, we get:

(2(1) - 1) + (2(2) - 1) + ... + (2(10) - 1)

The simplification of which is as follows:

1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 = 100

Example 5:

Find the sum of Σ (from i=1 to 6) of (2i² + 3i).

Expanding, we calculate each term:

For i=1: 2(1)² + 3(1) = 5
For i=2: 2(2)² + 3(2) = 14
For i=3: 2(3)² + 3(3) = 27
For i=4: 2(4)² + 3(4) = 44
For i=5: 2(5)² + 3(5) = 65
For i=6: 2(6)² + 3(6) = 90

Adding up these results:

5 + 14 + 27 + 44 + 65 + 90 = 245

Conclusion

Sigma notation is a simple but powerful mathematical tool that provides a concise way to write long sums. Whether you are working with arithmetic or geometric sequences or dealing with more complex formulas, sigma notation helps to represent these sums concisely. By understanding how to manipulate and interpret sigma notation, you can solve complex series and sequences more easily. Mastery of this notation is also foundational to advanced mathematics, statistics, and various fields of science.

As you continue to practice and work with sigma notation, remember to break down expressions, recognize limits and functions, and always check each term. By doing so, you will develop a deeper understanding of sequences and series, which will be invaluable in your mathematical journey.


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