Arithmetic and Geometric Sequences and Series

Arithmetic sequences

Each term adds a common difference d.

• uₙ = u₁ + (n − 1)d
• Sₙ = (n/2)(2u₁ + (n − 1)d) = (n/2)(u₁ + uₙ)

Example. 3, 7, 11, ...: u₂₀ = 3 + 19(4) = 79; S₂₀ = 10(3 + 79) = 820.

Geometric sequences

Each term is multiplied by a common ratio r.

• uₙ = u₁ rⁿ⁻¹
• Sₙ = u₁(rⁿ − 1)/(r − 1), r ≠ 1

Example. 2, 6, 18, ...: u₆ = 2 × 3⁵ = 486; S₆ = 2(729 − 1)/2 = 728.

Infinite geometric series

If abs(r) < 1: S∞ = u₁ / (1 − r).

8 + 4 + 2 + ... = 8/(1 − 1/2) = 16.

Sigma notation

Σ (k=1 to 5) (2k + 1) = 3 + 5 + 7 + 9 + 11 = 35.

Financial applications

• Compound interest: FV = PV(1 + r/(100k))^(kn), with k compounding periods per year.
• Depreciation: value × (1 − r/100)ⁿ.
• Use the TVM solver on the GDC for loans and annuities.

Example. $5000 at 4% per year compounded monthly for 3 years: 5000(1 + 0.04/12)³⁶ ≈ $5636.36.

Quick summary

• Arithmetic: add d; geometric: multiply by r.
• S∞ = u₁/(1 − r) only when abs(r) < 1.
• Compound interest is a geometric sequence.