Geometric progression - practice problems

A geometric progression (or geometric sequence) is a sequence where each term after the first is found by multiplying the previous term by a fixed non-zero number called the common ratio. The general form is a, ar, ar², ar³, ... where a is the first term and r is the common ratio. The nth term is given by aₙ = a·r^(n-1). The sum of the first n terms is Sₙ = a(1-r^n)/(1-r) when r ≠ 1. Geometric progressions model exponential growth and decay, compound interest, and various scientific phenomena. When |r| < 1, an infinite geometric series can converge to a finite sum. Understanding geometric progressions is essential for sequences, series, and mathematical modeling.