Geometric Sequences

A geometric sequence multiplies by a fixed amount, the common ratio r, each step.

3, 6, 12, 24, … has r = 2.
81, 27, 9, 3, … has r = 1/3.

Any term is given by:
aₙ = a₁·r^(n−1)

The 8th term of 3, 6, 12, …: a₈ = 3·2⁷ = 384.

Find r by dividing any term by the one before it. Division, not subtraction — that is the whole distinction from arithmetic.

A negative ratio makes the signs alternate: 2, −6, 18, −54 has r = −3.

A geometric sequence satisfies aₙ = r·aₙ₋₁ with explicit formula aₙ = a₁r^{n−1}. It is an exponential function on a discrete domain, just as an arithmetic sequence is a linear one, and the same n − 1 offset appears for the same reason.

The finite sum is Sₙ = a₁(1 − rⁿ)/(1 − r) for r ≠ 1, derived by subtracting rSₙ from Sₙ so that all interior terms cancel — the same telescoping trick used to convert repeating decimals to fractions.

When |r| < 1, rⁿ → 0 and the infinite series converges to S = a₁/(1 − r). This is why 0.999… = 1: it is the geometric series with a₁ = 0.9 and r = 0.1. For |r| ≥ 1 the series diverges.

Distinguishing sequence types is a matter of testing differences and ratios: constant differences mean arithmetic, constant ratios mean geometric, and neither means some other pattern entirely.