Sequence and Series

 

Sequence and Series

Sequence

• Definition: A sequence is an ordered list of numbers or objects.
• Types:
  • Arithmetic Sequence: Each term differs from the preceding term by a constant amount (common difference).
  • Geometric Sequence: Each term is a constant multiple of the preceding term (common ratio).
  • Harmonic Sequence: The reciprocals of the terms form an arithmetic sequence.
• General Term: The nth term of a sequence is denoted by aₙ.
• Finite Sequence: A sequence with a definite number of terms.
• Infinite Sequence: A sequence with an unlimited number of terms.

Series

• Definition: A series is the sum of the terms of a sequence.
• Types:
  • Arithmetic Series: The sum of an arithmetic sequence.
  • Geometric Series: The sum of a geometric sequence.
  • Harmonic Series: The sum of a harmonic sequence.
• Finite Series: The sum of a finite number of terms.
• Infinite Series: The sum of an infinite number of terms.

Important Formulas

• Arithmetic Sequence:
  • aₙ = a₁ + (n-1)d (where a₁ is the first term, d is the common difference, and n is the term number)
  • Sₙ = n/2 [2a₁ + (n-1)d] (where Sₙ is the sum of the first n terms)
• Geometric Sequence:
  • aₙ = a₁r^(n-1) (where a₁ is the first term, r is the common ratio, and n is the term number)
  • Sₙ = a₁(1-r^n)/(1-r) (where Sₙ is the sum of the first n terms)
• Infinite Geometric Series:
  • S = a₁/(1-r) (if |r| < 1)

Applications

• Finance: Compound interest, annuities, amortization
• Physics: Projectile motion, oscillations
• Engineering: Structural analysis, signal processing
• Computer Science: Algorithms, data structures
• Mathematics: Calculus, number theory