{
 "source": "https://peabody.kids/flashcards/math-sequences-series",
 "id": "math-sequences-series",
 "title": "Sequences, Series and Patterns",
 "category": "math",
 "subject": "Math",
 "levels": [
  "6-8",
  "9-12"
 ],
 "description": "Arithmetic and geometric sequences, formulas for sums, and famous number patterns.",
 "tags": [
  "sequences",
  "series",
  "patterns"
 ],
 "frontLabel": "Question",
 "backLabel": "Answer",
 "cards": [
  [
   "Sequence",
   "An ordered list of numbers following a rule"
  ],
  [
   "Term of a sequence",
   "One number in the list"
  ],
  [
   "Arithmetic sequence",
   "Each term is found by adding the same number"
  ],
  [
   "Common difference",
   "The number added each time in an arithmetic sequence"
  ],
  [
   "Geometric sequence",
   "Each term is found by multiplying by the same number"
  ],
  [
   "Common ratio",
   "The number multiplied each time in a geometric sequence"
  ],
  [
   "Explicit formula, arithmetic",
   "aₙ = a₁ + (n − 1)d"
  ],
  [
   "Explicit formula, geometric",
   "aₙ = a₁ · rⁿ⁻¹"
  ],
  [
   "Recursive formula",
   "Gives each term from the term before it"
  ],
  [
   "Sum of an arithmetic series",
   "Sₙ = n(a₁ + aₙ)/2"
  ],
  [
   "Sum of a finite geometric series",
   "Sₙ = a₁(1 − rⁿ)/(1 − r)"
  ],
  [
   "Sum of an infinite geometric series",
   "S = a₁/(1 − r), when |r| < 1"
  ],
  [
   "Next term: 2, 4, 6, 8, ...",
   "10 (add 2)"
  ],
  [
   "Next term: 5, 10, 15, 20, ...",
   "25 (add 5)"
  ],
  [
   "Next term: 3, 6, 12, 24, ...",
   "48 (multiply by 2)"
  ],
  [
   "Next term: 1, 4, 9, 16, ...",
   "25 (squares)"
  ],
  [
   "Next term: 1, 3, 6, 10, ...",
   "15 (triangular numbers, add 5)"
  ],
  [
   "Next term: 100, 90, 80, ...",
   "70 (subtract 10)"
  ],
  [
   "Next term: 81, 27, 9, 3, ...",
   "1 (divide by 3)"
  ],
  [
   "Next term: 1, 1, 2, 3, 5, 8, ...",
   "13 (Fibonacci)"
  ],
  [
   "Next term: 2, 6, 18, 54, ...",
   "162 (multiply by 3)"
  ],
  [
   "Next term: 1, 2, 4, 7, 11, ...",
   "16 (differences grow by 1)"
  ],
  [
   "Next term: 1, 8, 27, 64, ...",
   "125 (cubes)"
  ],
  [
   "Next term: 2, 3, 5, 7, 11, ...",
   "13 (primes)"
  ],
  [
   "10th term of 3, 7, 11, ...",
   "39"
  ],
  [
   "20th term of 5, 8, 11, ...",
   "62"
  ],
  [
   "8th term of 2, 4, 8, ...",
   "256"
  ],
  [
   "Common difference of 7, 12, 17, ...",
   "5"
  ],
  [
   "Common ratio of 5, 15, 45, ...",
   "3"
  ],
  [
   "Common ratio of 16, 8, 4, ...",
   "1/2"
  ],
  [
   "Sum of 1 + 2 + ... + 10",
   "55"
  ],
  [
   "Sum of 1 + 2 + ... + 100",
   "5,050"
  ],
  [
   "Sum of the first 5 odd numbers",
   "25"
  ],
  [
   "Sum of the first 10 even numbers",
   "110"
  ],
  [
   "Sum of 1 + 1/2 + 1/4 + 1/8 + ... forever",
   "2"
  ],
  [
   "Fibonacci rule",
   "Each term is the sum of the two before it"
  ],
  [
   "Fibonacci numbers up to 144",
   "1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144"
  ],
  [
   "Ratio of consecutive Fibonacci numbers",
   "Approaches the golden ratio, about 1.618"
  ],
  [
   "Triangular numbers formula",
   "n(n + 1)/2"
  ],
  [
   "Square numbers formula",
   "n²"
  ],
  [
   "Pentagonal number idea",
   "n(3n − 1)/2: 1, 5, 12, 22, ..."
  ],
  [
   "Pascal's triangle rows",
   "1; 1 1; 1 2 1; 1 3 3 1; 1 4 6 4 1"
  ],
  [
   "Row sums of Pascal's triangle",
   "Powers of 2"
  ],
  [
   "Sigma notation",
   "Σ means add up the terms for the given range"
  ],
  [
   "Σ from k=1 to 4 of k",
   "10"
  ],
  [
   "Σ from k=1 to 3 of k²",
   "14"
  ],
  [
   "Σ from k=1 to 5 of 2",
   "10"
  ],
  [
   "Pattern",
   "Something that repeats or changes by a rule"
  ],
  [
   "Growth pattern",
   "A pattern in which a number or shape grows by a rule each step"
  ],
  [
   "Limit of a sequence",
   "The value the terms get close to"
  ],
  [
   "Converge",
   "The terms or sum settle toward a number"
  ],
  [
   "Diverge",
   "The terms or sum do not settle toward a number"
  ],
  [
   "Harmonic series",
   "1 + 1/2 + 1/3 + 1/4 + ... ; it grows without bound"
  ],
  [
   "Arithmetic mean of two numbers",
   "Their average"
  ],
  [
   "Geometric mean of two numbers",
   "The square root of their product"
  ],
  [
   "Geometric mean of 4 and 9",
   "6"
  ],
  [
   "Compound interest as a sequence",
   "A geometric sequence, multiplying by (1 + r) each period"
  ]
 ]
}