Two of the most important sequences in algebra — known as AP (arithmetic progression) and GP (geometric progression / GP series) in UK/Indian curricula. Arithmetic sequences grow by adding a fixed amount (the common difference d); geometric sequences (GP) grow by multiplying by a fixed factor (the common ratio r). The nth term and partial sum each have their own clean closed form.

The Formulas

Name	Formula	Notes
Arithmetic — nth Term	`aₙ = a₁ + (n − 1)·d`	a₁ = first term, d = common difference (constant difference between consecutive terms).
Arithmetic — Sum of n Terms	`Sₙ = n/2 × (a₁ + aₙ) = n/2 × [2a₁ + (n−1)d]`	Two equivalent forms: average of endpoints × count, or expanded.
Geometric — nth Term	`aₙ = a₁ × rⁿ⁻¹`	a₁ = first term, r = common ratio (constant ratio between consecutive terms).
Geometric — Sum of n Terms	`Sₙ = a₁ × (1 − rⁿ) / (1 − r)`	r ≠ 1. When r = 1 the sequence is constant and Sₙ = n·a₁.
Geometric — Infinite Sum	`S∞ = a₁ / (1 − r)`	Converges only when \|r\| < 1; otherwise the sum diverges.
Common Difference	`d = aₙ₊₁ − aₙ`	How to detect arithmetic — same difference between every consecutive pair.
Common Ratio	`r = aₙ₊₁ / aₙ`	How to detect geometric — same ratio between every consecutive pair.

Worked Examples

Example 1: Arithmetic: 3, 7, 11, 15, … find a₁₀ and S₁₀

a₁ = 3, d = 4
a₁₀ = 3 + (10 − 1) × 4 = 3 + 36 = 39
S₁₀ = 10/2 × (3 + 39) = 5 × 42 = 210

Example 2: Geometric: 2, 6, 18, 54, … find a₈ and S₈

a₁ = 2, r = 3
a₈ = 2 × 3⁷ = 2 × 2187 = 4374
S₈ = 2 × (1 − 3⁸) / (1 − 3) = 2 × (1 − 6561) / (−2) = 6560

Example 3: Convergent infinite geometric: 1 + ½ + ¼ + ⅛ + …

a₁ = 1, r = 0.5 (|r| < 1, so it converges)
S∞ = 1 / (1 − 0.5) = 1 / 0.5 = 2

Example 4: Distinguish: is 5, 10, 20, 40 arithmetic or geometric?

Differences: 5, 10, 20 — not constant → not arithmetic
Ratios: 10/5 = 2, 20/10 = 2, 40/20 = 2 — constant → geometric, r = 2

Arithmetic & Geometric Sequence Formulas