Sequences as Models

Mathematical Modeling · Axiom Academy

Discover how ordered lists of numbers can capture the essence of change that happens step by step. Explore what happens when we list values one after another. Click the terms to see how sequences are indexed. 2. Two Ways to Define a Sequence There are two fundamental approaches to describing how a sequence works. Explore both methods below. Watch how a simple savings account grows year by year. Adjust the interest rate to see compound growth in action. See how a population changes generation by generation. Each organism produces offspring, and some survive to the next generation. 5. The Bridge to Difference Equations See how a recursive sequence formula transforms into a difference equation - the language of discrete dynamics. Create a custom sequence model by choosing the type and parameters. Watch how small changes affect long-term behavior. Sequences model processes that change in distinct steps: yearly account balances, daily temperatures, monthly sales, or generational populations. Each index n represents a moment in discrete time. Explicit formulas give direct answers. Recursive formulas reveal underlying mechanisms. Both perspectives are valuable - the best modelers can switch between them fluidly. Adding a constant gives arithmetic sequences (linear change). Multiplying by a constant gives geometric sequences (exponential change). These are the building blocks of discrete modeling.

This is the written version of the interactive lesson above. See the full Mathematical Modeling course.