Assess students' ability to create and analyze a family of integer sequences defined by nonlinear recurrences: require them to construct sequences (parameterized by an integer parameter) satisfying a given nonlinear recurrence, prove the family attains a specified exponential growth rate (target base r>1) by providing upper and lower bounds, and establish prescribed divisibility properties (e.g., term a_n divisible by a given function m(n) or by earlier terms). The task tests recurrence design, induction, inequality/bounding techniques, modular arithmetic, and clear proof writing; include explicit examples and complete proofs for general parameter values.