Sequences in Mathematics

Sequences

Definition

• A sequence is an arrangement of objects, numbers, or events in a specific order.
• It can be finite (having a fixed number of terms) or infinite (having an unlimited number of terms).

Types of Sequences

• Finite sequence: A sequence with a fixed number of terms.
• Infinite sequence: A sequence with an unlimited number of terms.
• Arithmetic sequence (AP): A sequence with a constant difference between consecutive terms.
• Geometric sequence (GP): A sequence with a constant ratio between consecutive terms.

Arithmetic Sequence (AP)

• Formula: an = a1 + (n - 1)d, where an is the nth term, a1 is the first term, and d is the common difference.
• Properties:
  • The sum of the first n terms: Sn = (n/2)(2a1 + (n - 1)d)
  • The nth term from the end: an' = a1 + (n - 1)(-d)

Geometric Sequence (GP)

• Formula: an = ar^(n - 1), where an is the nth term, a is the first term, and r is the common ratio.
• Properties:
  • The sum of the first n terms: Sn = (a(1 - r^n))/(1 - r)
  • The nth term from the end: an' = a/r^(n - 1)

nth Term and Sum of a Sequence

• nth term: The formula to find the nth term of a sequence.
• Sum of a sequence: The formula to find the sum of the first n terms of a sequence.

Examples and Applications

• Real-world applications of sequences include population growth, financial calculations, and data analysis.
• Examples of sequences include:
  • 2, 5, 8, 11, ... (arithmetic sequence)
  • 2, 6, 18, 34, ... (geometric sequence)

Sequences

• A sequence is an arrangement of objects, numbers, or events in a specific order, which can be finite or infinite.

Types of Sequences

• Finite sequence: A sequence with a fixed number of terms.
• Infinite sequence: A sequence with an unlimited number of terms.
• Arithmetic sequence (AP): A sequence with a constant difference between consecutive terms.
• Geometric sequence (GP): A sequence with a constant ratio between consecutive terms.

Arithmetic Sequence (AP)

• Formula: an = a1 + (n - 1)d, where an is the nth term, a1 is the first term, and d is the common difference.
• Properties:
  • Sum of the first n terms: Sn = (n/2)(2a1 + (n - 1)d)
  • nth term from the end: an' = a1 + (n - 1)(-d)

Geometric Sequence (GP)

• Formula: an = ar^(n - 1), where an is the nth term, a is the first term, and r is the common ratio.
• Properties:
  • Sum of the first n terms: Sn = (a(1 - r^n))/(1 - r)
  • nth term from the end: an' = a/r^(n - 1)

nth Term and Sum of a Sequence

• nth term: The formula to find the nth term of a sequence.
• Sum of a sequence: The formula to find the sum of the first n terms of a sequence.

Examples and Applications

• Real-world applications of sequences include population growth, financial calculations, and data analysis.
• Examples of sequences include:
  • 2, 5, 8, 11,... (arithmetic sequence)
  • 2, 6, 18, 34,... (geometric sequence)

Sets

• A set is a collection of unique objects, known as elements or members.
• Sets are denoted by capital letters (e.g. A, B, C).
• Elements are denoted by small letters (e.g. a, b, c).
• Set operations include:
  • Union (A ∪ B): combining elements of A and B.
  • Intersection (A ∩ B): finding common elements of A and B.
  • Difference (A - B): finding elements in A but not in B.
  • Complement (A'): finding elements not in A.

Relations and Functions

• A relation is a set of ordered pairs (a, b) where a and b are elements of two sets.
• The domain is the set of all first elements (a) in the ordered pairs.
• The range is the set of all second elements (b) in the ordered pairs.
• A function is a relation where every element in the domain corresponds to exactly one element in the range.
• Types of functions include:
  • Injective (one-to-one): every element in the range corresponds to exactly one element in the domain.
  • Surjective (onto): every element in the range is mapped to at least one element in the domain.
  • Bijective (one-to-one and onto): both injective and surjective.

Trigonometry

• Angles are measured by the rotation from the initial side to the terminal side.
• Trigonometric ratios include:
  • Sine (sin): opposite side over hypotenuse.
  • Cosine (cos): adjacent side over hypotenuse.
  • Tangent (tan): opposite side over adjacent side.
  • Cotangent (cot): adjacent side over opposite side.
  • Secant (sec): hypotenuse over adjacent side.
  • Cosecant (cosec): hypotenuse over opposite side.
• Important trigonometric identities include:
  • sin^2(A) + cos^2(A) = 1.
  • tan(A) = sin(A) / cos(A).

Complex Numbers

• Complex numbers are numbers of the form a + bi, where a and b are real numbers and i = √(-1).
• Properties of complex numbers include:
  • i^2 = -1.
  • i^3 = -i.
  • i^4 = 1.
• Operations with complex numbers include:
  • Addition: (a + bi) + (c + di) = (a + c) + (b + d)i.
  • Subtraction: (a + bi) - (c + di) = (a - c) + (b - d)i.
  • Multiplication: (a + bi) × (c + di) = (ac - bd) + (ad + bc)i.
• The conjugate of a + bi is a - bi.
• The modulus of a complex number is |a + bi| = √(a^2 + b^2).

Statistics and Probability

• Statistics is the study of collection, analysis, and interpretation of data.
• Probability is a measure of the likelihood of an event occurring.
• Types of data include:
  • Qualitative: categorical data.
  • Quantitative: numerical data.
• Measures of central tendency include:
  • Mean.
  • Median.
  • Mode.
• Measures of dispersion include:
  • Range.
  • Variance.
  • Standard deviation.
• The probability of an event is P(E) = Number of favorable outcomes / Total number of outcomes.