Classify formulas and sequences

Learning Objectives Distinguish between arithmetic and geometric sequences by analyzing their common difference or common ratio. Identify a sequence as neither arithmetic nor geometric if it lacks a constant difference or ratio. Classify a formula for a sequence as either explicit or recursive. Write the first few terms of a sequence given its explicit formula. Write the first few terms of a sequence given its recursive formula and initial term(s). Determine the type of sequence (arithmetic, geometric, or neither) from a given list of terms. Ever notice how a sunflower's seeds spiral outwards or how your savings grow with interest? 🌱 These are real-life sequences! But how do we describe their patterns mathematically? In this tutorial, we will learn how to identify and...

TermDefinitionExample SequenceAn ordered list of numbers, called terms, that follow a specific pattern or rule.The list 2, 4, 6, 8, 10, ... is a sequence. Arithmetic SequenceA sequence where the difference between any two consecutive terms is constant. This constant value is called the common difference (d).In the sequence 5, 8, 11, 14, ..., the common difference is d = 3. Geometric SequenceA sequence where the ratio of any two consecutive terms is constant. This constant value is called the common ratio (r).In the sequence 2, 6, 18, 54, ..., the common ratio is r = 3. Explicit FormulaA formula that defines the nth term of a sequence, a_n, as a function of its position, n. It allows for the direct calculation of any term in the sequence.a_n = 2n + 1. To find the 10th term, you just plug i...

Explicit Formula for an Arithmetic Sequence a_n = a_1 + (n-1)d Use this to find any term (a_n) in an arithmetic sequence when you know the first term (a_1) and the common difference (d). Recursive Formula for an Arithmetic Sequence a_n = a_{n-1} + d, given a_1 Use this to define a term based on the previous term. You must state the first term, a_1, to provide a starting point. Explicit Formula for a Geometric Sequence a_n = a_1 * r^(n-1) Use this to find any term (a_n) in a geometric sequence when you know the first term (a_1) and the common ratio (r). Recursive Formula for a Geometric Sequence a_n = a_{n-1} * r, given a_1 Use this to define a term based on the previous term. You must state the first term, a_1, to provide a starting point.