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 ↓


Arithmetic Sequence


Introduction to sequence

In mathematics, a sequence is a group of numbers arranged in a specific order. Each number in the sequence is called a term. Sequences can follow patterns, making it easier for us to predict subsequent numbers or describe the sequence with a formula.

What is an arithmetic sequence?

An arithmetic sequence is a type of sequence in which the difference between consecutive terms is constant. This difference is called the "common difference." When you know the common difference and at least one term in the sequence, you can easily find the other terms.

Normal form of an arithmetic sequence

The arithmetic sequence can be described by the formula:

a_n = a_1 + (n - 1) cdot d

Where:

  • a_n is the n -th term of the sequence
  • a_1 is the first term of the sequence
  • d is the common difference
  • n is the position of the term in the sequence

Common difference

The common difference is the center of arithmetic sequences. It is the value that is continuously added (or subtracted) to get the next term from the previous term.

If an arithmetic sequence has the following terms: a_1, a_2, a_3, ..., then the common difference d can be calculated as:

d = a_2 - a_1 = a_3 - a_2 = ...

Example

Consider the sequence: 5, 8, 11, 14, 17, ...

Here, the common difference d is 3 because:

8 - 5 = 3, 11 - 8 = 3, 14 - 11 = 3, 17 - 14 = 3

Thus, with a common difference of 3, 5 is followed by 8, 11, 14, and so on.

Finding a specific term

To find any term in an arithmetic sequence, we can use the formula mentioned above. Let us see how to find a specific term with an example.

Example

Suppose we have an arithmetic sequence where the first term a_1 = 3 and the common difference d = 5. Let us find the 10th term. 
a_{10} = a_1 + (10-1) cdot d = 3 + 9 cdot 5 = 3 + 45 = 48

Therefore, the 10th term is a_{10} 48.

Visual representation of arithmetic sequences

A_1 A_2 a_3 a_4 a_5 a_6

The line given above represents an arithmetic sequence where the difference between consecutive circles is the common difference. Each circle represents a term.

Sum of an arithmetic sequence

In many scenarios, we may need to find the sum of an arithmetic sequence. To achieve this, we use the formula for the sum of the first n terms of an arithmetic sequence, which is:

S_n = (n / 2) cdot (a_1 + a_n)

Alternatively, replace a_n with the formula for the n -th term:

S_n = (n / 2) cdot [2a_1 + (n - 1) cdot d]

Example

Continuing from the previous example, find the sum of the first 10 terms where a_1 = 3 and d = 5. 
n = 10
a_1 = 3
First, find a_{10}:
a_{10} = a_1 + 9 cdot 5 = 48
Now find S_{10}:
S_{10} = (10 / 2) cdot (3 + 48) = 5 cdot 51 = 255

Thus, the sum of the first 10 terms, S_{10}, is 255.

Arithmetic sequences in real life

Arithmetic sequences are not only theoretical constructions, but can also appear in practical situations. Here are some real-life examples:

  • Monthly savings where the same amount is saved every month.
  • Depositing a fixed sum of money regularly in a bank account.
  • A clock that moves forward a few seconds every day, creating a predictable pattern.
  • Construction phases where each phase requires the same number of more steps than the previous one.

Conclusion

Arithmetic sequences are an important concept in algebra, providing a basis for understanding more complex sequences and series. With the knowledge of the first term and the common difference, you can create a complete sequence and find any term or sum you want.

By understanding this fundamental concept, you open the doors to explore deeper mathematical ideas and practical applications, making arithmetic sequences an important part of mathematics.


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