Sequences and Series Flashcards

Grade 9 · Math · Flashcards

Master every key formula and concept for arithmetic and geometric sequences and their sums.

Flashcards

Arithmetic sequence definition: An ordered list where consecutive terms differ by a constant amount d (common difference)
Geometric sequence definition: An ordered list where consecutive terms have a constant ratio r (common ratio)
Arithmetic explicit formula: aₙ = a₁ + (n − 1)d
Geometric explicit formula: aₙ = a₁ · r^(n−1)
Arithmetic recursive formula: aₙ = aₙ₋₁ + d, with a₁ given
Geometric recursive formula: aₙ = r · aₙ₋₁, with a₁ given
Arithmetic series sum formula 1: Sₙ = n/2 · (a₁ + aₙ) — use when first and last terms are known
Arithmetic series sum formula 2: Sₙ = n/2 · [2a₁ + (n − 1)d] — use when last term is not given
Geometric series sum formula: Sₙ = a₁(1 − rⁿ)/(1 − r) for r ≠ 1
Sigma notation Σ: Σ means 'sum of'; Σᵢ₌₁ⁿ aᵢ means add terms a₁ through aₙ
Identify sequence type from table: Check differences (arithmetic) then ratios (geometric). If neither is constant, it is neither type
Find a₁ given aₙ = 50, d = 4, n = 13: 50 = a₁ + 12·4 = a₁ + 48, so a₁ = 2
Sum of 1 + 1 + 1 + ... (n terms): n (this is an arithmetic series with d = 0)
Geometric series with r = 1: Sum = n · a₁ (each term is the same, just multiply by the number of terms)