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

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