Sequences & Series

Arithmetic and geometric sequences and series

Arithmetic and Geometric Sequences and Series

An arithmetic sequence has a common difference (d): Tₙ = a + (n−1)d. A geometric sequence has a common ratio (r): Tₙ = a·rⁿ⁻¹. A series is the sum of terms: Sₙ = n/2[2a + (n−1)d] for arithmetic, Sₙ = a(rⁿ−1)/(r−1) for geometric.

Example

Arithmetic Sequence

Sequence: 5, 8, 11, 14, ... a = 5, d = 3 T₁₀ = 5 + (10−1)(3) = 5 + 27 = 32 S₁₀ = 10/2[2(5) + 9(3)] = 5[10 + 27] = 185

Example

Geometric Sequence

Sequence: 2, 6, 18, 54, ... a = 2, r = 3 T₆ = 2(3)⁵ = 2(243) = 486 S₆ = 2(3⁶ − 1)/(3 − 1) = 2(728)/2 = 728

Note

Remember

Arithmetic: constant difference between terms, linear growth. Geometric: constant ratio between terms, exponential growth. Check: Tₙ − Tₙ₋₁ = constant (arithmetic) or Tₙ/Tₙ₋₁ = constant (geometric).

Key Vocabulary

Arithmetic sequenceA sequence with a constant difference between consecutive terms

Geometric sequenceA sequence with a constant ratio between consecutive terms

Common differenceThe fixed amount added each time (d)

Common ratioThe fixed multiplier between terms (r)

SeriesThe sum of terms in a sequence

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Arithmetic sequence

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