Geometric progression
A sequence with first term b₁ ≠ 0 in which each term from the second is the previous term multiplied by the same number q ≠ 0 is called a geometric progression: b_(n+1) = b_n · q. The number q is the common ratio: q = b_(n+1) : b_n. For example, in 3, 6, 12, 24, ... q = 2; in 8, 4, 2, 1, ... q = 1/2; in 5, −10, 20, ... q = −2. To prove that a sequence is a geometric progression we show that b_(n+1) : b_n does not depend on n. The n-th term formula is b_n = b₁ · q^(n−1). In a progression with all terms positive, b_n² = b_(n−1) · b_(n+1), i.e. each term is the geometric mean of its neighbours: b_n = √(b_(n−1) b_(n+1)); this gives the name. If two terms are given, dividing them gives a power of q.
“Multiplier chain”: one student states b₁ and q, the class takes turns computing b₂, b₃, b₄; then the reverse: terms are given and q must be found.