Arithmetic Sequence

Each term is found by adding a constant difference (d) to the previous term.

nth term: aₙ = a₁ + (n - 1)d

Example: 3, 7, 11, 15, ... (d = 4)

Arithmetic Series Sum

Sₙ = n(a₁ + aₙ) / 2

Or equivalently: Sₙ = n[2a₁ + (n-1)d] / 2

Geometric Sequence

Each term is found by multiplying the previous term by a constant ratio (r).

nth term: aₙ = a₁ × r⁽ⁿ⁻¹⁾

Example: 2, 6, 18, 54, ... (r = 3)

Geometric Series Sum

• Finite (n terms): Sₙ = a₁(1 - rⁿ) / (1 - r), when r ≠ 1
• Infinite (|r| < 1): S = a₁ / (1 - r)

Fibonacci Sequence

Each term is the sum of the two preceding ones: 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

aₙ = aₙ₋₁ + aₙ₋₂

Square Numbers

Perfect squares of natural numbers: 1, 4, 9, 16, 25, 36, 49, ...

aₙ = n²

Convergence and Divergence

• Convergent: An infinite series converges if the sum approaches a finite value. Example: 1 + 1/2 + 1/4 + 1/8 + ... = 2
• Divergent: An infinite series diverges if the sum grows without bound or oscillates. Example: 1 + 2 + 3 + 4 + ... = ∞

Finance

Compound interest, loan payments, and annuities use geometric sequences and series.

Computer Science

Algorithm analysis, recursion, and data structure capacity calculations.

Physics

Model wave patterns, decay processes, and harmonic motion using series.

Nature

Fibonacci appears in flower petals, pine cones, and spiral galaxies.