(˶˃ ᵕ ˂˶).ᐟ

Sequences

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Applications

  • Musical sequences → chord progression
  • Programming → iteration
  • Finance → interest

Concepts

  • Similar yet different to sets

Order Matters

  • Ordered, unlike sets
  • {1,2,3}={3,2,1}
  • (1,2,3)≠(3,2,1)

Finite vs Infinite Sequences

  • Finite → days of the week, alphabet
  • Infinite → natural numbers

Index Notation

  • Sequences start at a certain indexes depending on the context
  • 𝑎1,𝑎2,𝑎3 or 𝑎0,𝑎1,𝑎2

Examples

  • 2,4,6,8→𝑎𝑛=2𝑛where𝑛≥1
    • 𝑎1=2⋅1→2
  • 1,12,14,18→𝑎𝑛=(12)𝑛−1where𝑛≥1
    • 𝑎1=(12)1−1→(12)0→1
  • 0,3,6,9→𝑎𝑛=3(𝑛−1)where𝑛≥1
    • 𝑎1=3(1−1)→3(0)→0

Explicit vs Recursive

  • Recursive → using previous values of the sequence
  • Explicit → 𝑎𝑛=2𝑛
  • Recursive → 𝑎𝑛=𝑎𝑛−1+𝑎𝑛−2

Examples

  • Given 𝑎𝑛=𝑎𝑛−1+3 and 𝑎1=2, find 𝑎5
    • 𝑎1=2
    • 𝑎2=𝑎1+3→2+3→5
    • 𝑎3=𝑎2+3→5+3→8
    • 𝑎4=𝑎3+3→8+3→11
    • 𝑎5=𝑎4+3→11+3→14
  • Given 𝑎=𝑎𝑛−1(−1) and 𝑎1=5, find 𝑎4
    • 𝑎1=5
    • 𝑎2=𝑎1(−1)→5(−1)→−5
    • 𝑎3=𝑎2(−1)→−5(−1)→5
    • 𝑎4=𝑎3(−1)→5(−1)→−5

Arithmetic vs Geometric

Arithmetic

  • Linear, constant difference → 𝑑
    • 3,7,11,15→𝑑=4
  • General form → 𝑎𝑛=𝑎𝑛−1+1

Geometric

  • Quadratic, constant ratio → 𝑟
    • 2,4,6,8→𝑟=2
  • General form → 𝑎𝑛=𝑎𝑛−1𝑟𝑛−1

Summation

  • Denoted with ∑𝑏𝑖=𝑎𝑐
    • 𝑖 → index
    • 𝑎 → start value
    • 𝑏 → end value
    • 𝑐 → body

Example

  • ∑4𝑖=12𝑖
    • 𝑖=1:2(1)→2
    • 𝑖=2:2(2)→4
    • 𝑖=3:2(3)→6
    • 𝑖=4:2(4)→8
    • →2+4+6+8=20

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