Convert between explicit and recursive formulas

Learning Objectives Define and differentiate between explicit and recursive formulas for sequences. Convert a given explicit formula for an arithmetic sequence into its recursive form. Convert a given explicit formula for a geometric sequence into its recursive form. Convert a given recursive formula for an arithmetic or geometric sequence into its explicit form. Analyze the structure of a formula to determine if a sequence is arithmetic, geometric, or neither. Relate the explicit formula of a sequence to its long-term behavior and determine its limit. How does a bank calculate your loan interest month after month, or a computer program render a complex fractal? 🤔 It's all about sequences, defined by formulas that we can switch between! This tutorial will teach you ho...

TermDefinitionExample SequenceAn ordered list of numbers, where each number is called a term. We denote the nth term as a_n.The sequence of even positive integers is 2, 4, 6, 8, ... Explicit FormulaA rule that defines the nth term (a_n) of a sequence as a direct function of its position, n. It allows you to calculate any term without knowing the previous one.For the sequence 2, 4, 6, 8, ..., the explicit formula is a_n = 2n. To find the 50th term, you just calculate a_50 = 2(50) = 100. Recursive FormulaA rule that defines a term based on one or more preceding terms. It always requires a starting value, known as the base case (usually a_1).For the sequence 2, 4, 6, 8, ..., the recursive formula is a_1 = 2 and a_n = a_{n-1} + 2 for n > 1. Arithmetic SequenceA sequence where the differenc...

Arithmetic Sequence Formulas Explicit: a_n = a_1 + (n-1)d | Recursive: a_1 = [initial value], a_n = a_{n-1} + d Use for sequences with a common difference, d. The explicit form is a linear function of n. An arithmetic sequence only converges if d=0; otherwise, it diverges to ±∞. Geometric Sequence Formulas Explicit: a_n = a_1 * r^(n-1) | Recursive: a_1 = [initial value], a_n = a_{n-1} * r Use for sequences with a common ratio, r. The explicit form is an exponential function of n. This is the most important form for limit calculations. Limit of a Geometric Sequence Given a_n = a_1 * r^(n-1), the limit lim_{n->∞} a_n is 0 if |r| < 1, a_1 if r = 1, and diverges if |r| > 1 or r = -1. This rule directly connects the explicit formula's common ratio to the seq...