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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

  • 2,4,6,8𝑎𝑛=2𝑛where𝑛1
    • 𝑎1=212
  • 1,12,14,18𝑎𝑛=(12)𝑛1where𝑛1
    • 𝑎1=(12)11(12)01
  • 0,3,6,9𝑎𝑛=3(𝑛1)where𝑛1
    • 𝑎1=3(11)3(0)0

Explicit vs Recursive

  • Recursive → using previous values of the sequence
  • Explicit → 𝑎𝑛=2𝑛
  • Recursive → 𝑎𝑛=𝑎𝑛1+𝑎𝑛2

Examples

  • Given 𝑎𝑛=𝑎𝑛1+3 and 𝑎1=2, find 𝑎5
    • 𝑎1=2
    • 𝑎2=𝑎1+32+35
    • 𝑎3=𝑎2+35+38
    • 𝑎4=𝑎3+38+311
    • 𝑎5=𝑎4+311+314
  • Given 𝑎=𝑎𝑛1(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
    • 3,7,11,15𝑑=4
  • General form → 𝑎𝑛=𝑎𝑛1+1

Geometric

  • Quadratic, constant ratio → 𝑟
    • 2,4,6,8𝑟=2
  • General form → 𝑎𝑛=𝑎𝑛1𝑟𝑛1

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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