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Sequences, Series and Patterns
Arithmetic and geometric sequences, formulas for sums, and famous number patterns.
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57 cards
- SequenceQuestionAn ordered list of numbers following a rule
- Term of a sequenceQuestionOne number in the list
- Arithmetic sequenceQuestionEach term is found by adding the same number
- Common differenceQuestionThe number added each time in an arithmetic sequence
- Geometric sequenceQuestionEach term is found by multiplying by the same number
- Common ratioQuestionThe number multiplied each time in a geometric sequence
- Explicit formula, arithmeticQuestionaₙ = a₁ + (n − 1)d
- Explicit formula, geometricQuestionaₙ = a₁ · rⁿ⁻¹
- Recursive formulaQuestionGives each term from the term before it
- Sum of an arithmetic seriesQuestionSₙ = n(a₁ + aₙ)/2
- Sum of a finite geometric seriesQuestionSₙ = a₁(1 − rⁿ)/(1 − r)
- Sum of an infinite geometric seriesQuestionS = a₁/(1 − r), when |r| < 1
- Next term: 2, 4, 6, 8, ...Question10 (add 2)
- Next term: 5, 10, 15, 20, ...Question25 (add 5)
- Next term: 3, 6, 12, 24, ...Question48 (multiply by 2)
- Next term: 1, 4, 9, 16, ...Question25 (squares)
- Next term: 1, 3, 6, 10, ...Question15 (triangular numbers, add 5)
- Next term: 100, 90, 80, ...Question70 (subtract 10)
- Next term: 81, 27, 9, 3, ...Question1 (divide by 3)
- Next term: 1, 1, 2, 3, 5, 8, ...Question13 (Fibonacci)
- Next term: 2, 6, 18, 54, ...Question162 (multiply by 3)
- Next term: 1, 2, 4, 7, 11, ...Question16 (differences grow by 1)
- Next term: 1, 8, 27, 64, ...Question125 (cubes)
- Next term: 2, 3, 5, 7, 11, ...Question13 (primes)
- 10th term of 3, 7, 11, ...Question39
- 20th term of 5, 8, 11, ...Question62
- 8th term of 2, 4, 8, ...Question256
- Common difference of 7, 12, 17, ...Question5
- Common ratio of 5, 15, 45, ...Question3
- Common ratio of 16, 8, 4, ...Question1/2
- Sum of 1 + 2 + ... + 10Question55
- Sum of 1 + 2 + ... + 100Question5,050
- Sum of the first 5 odd numbersQuestion25
- Sum of the first 10 even numbersQuestion110
- Sum of 1 + 1/2 + 1/4 + 1/8 + ... foreverQuestion2
- Fibonacci ruleQuestionEach term is the sum of the two before it
- Fibonacci numbers up to 144Question1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144
- Ratio of consecutive Fibonacci numbersQuestionApproaches the golden ratio, about 1.618
- Triangular numbers formulaQuestionn(n + 1)/2
- Square numbers formulaQuestionn²
- Pentagonal number ideaQuestionn(3n − 1)/2: 1, 5, 12, 22, ...
- Pascal's triangle rowsQuestion1; 1 1; 1 2 1; 1 3 3 1; 1 4 6 4 1
- Row sums of Pascal's triangleQuestionPowers of 2
- Sigma notationQuestionΣ means add up the terms for the given range
- Σ from k=1 to 4 of kQuestion10
- Σ from k=1 to 3 of k²Question14
- Σ from k=1 to 5 of 2Question10
- PatternQuestionSomething that repeats or changes by a rule
- Growth patternQuestionA pattern in which a number or shape grows by a rule each step
- Limit of a sequenceQuestionThe value the terms get close to
- ConvergeQuestionThe terms or sum settle toward a number
- DivergeQuestionThe terms or sum do not settle toward a number
- Harmonic seriesQuestion1 + 1/2 + 1/3 + 1/4 + ... ; it grows without bound
- Arithmetic mean of two numbersQuestionTheir average
- Geometric mean of two numbersQuestionThe square root of their product
- Geometric mean of 4 and 9Question6
- Compound interest as a sequenceQuestionA geometric sequence, multiplying by (1 + r) each period
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