Evaluate recursive formulas for sequences

Learning Objectives Define a recursive formula and identify its components. Calculate the first 'n' terms of a sequence given a recursive formula and an initial term. Evaluate a specific term (e.g., the 6th term) of a sequence by applying the recursive rule iteratively. Distinguish between a recursive formula and an explicit formula for a sequence. Model simple real-world scenarios, like compound interest or population growth, using a recursive formula. Use a calculator to accurately compute terms in sequences with more complex recursive rules. Ever see a pattern in nature, like the spirals on a pinecone, and wonder how it's built? 🌲 Many of these patterns can be described by recursive formulas, where each step builds upon the last! This tutorial will teach...

TermDefinitionExample SequenceAn ordered list of numbers, often following a specific pattern or rule. Each number in the list is called a term.The sequence of even positive integers is 2, 4, 6, 8, 10, ... Recursive FormulaA formula that defines each term of a sequence by relating it to the preceding term(s). It consists of two parts: an initial term and a recursive rule.a_1 = 4 and a_n = a_{n-1} + 2. This means the first term is 4, and every subsequent term is 2 more than the one before it. Initial Term(s)The first term (or first few terms) of a sequence, which is given explicitly. This is the starting point for the sequence.In the formula a_1 = 10 and a_n = 2 * a_{n-1}, the initial term is a_1 = 10. Recursive RuleThe part of the formula that specifies how to calculate the nth term (a_n)...

General First-Order Recursive Formula Given a_1, a_n = f(a_{n-1}) for n > 1 This is the standard form where the next term (a_n) is some function of the immediately preceding term (a_{n-1}). To find any term, you must know the term before it. Recursive Formula for an Arithmetic Sequence Given a_1, a_n = a_{n-1} + d for n > 1 Used for sequences where a constant difference, 'd', is added to each term to get the next. 'd' is the common difference. Recursive Formula for a Geometric Sequence Given a_1, a_n = a_{n-1} * r for n > 1 Used for sequences where each term is multiplied by a constant ratio, 'r', to get the next. 'r' is the common ratio. General Second-Order Recursive Formula Given a_1 and a_2, a_n = f(a_{n-1},...