Use a rule to complete a number sequence

Learning Objectives Identify if a number sequence is arithmetic, geometric, or quadratic. Calculate the common difference or common ratio for a given sequence. Derive the algebraic rule (nth term formula) for linear (arithmetic) and geometric sequences. Derive the algebraic rule (nth term formula) for quadratic sequences using the method of finite differences. Use a derived rule to find any term in a sequence, including missing terms or future terms. Apply sequence rules to solve word problems involving patterns. Ever wonder how video games generate levels or how your savings grow over time? 🎮 It all comes down to predictable patterns called sequences! This tutorial will teach you how to decode the hidden rules behind lists of numbers. You will learn to identify different...

TermDefinitionExample Number SequenceAn ordered list of numbers, called terms, that follow a specific pattern or rule.The list 2, 5, 8, 11, ... is a number sequence. Term (a_n)A single number in a sequence. The subscript 'n' indicates the term's position (e.g., a_1 is the first term, a_5 is the fifth term).In the sequence 2, 5, 8, 11, ..., the third term (a_3) is 8. Arithmetic SequenceA sequence where the difference between consecutive terms is constant. This constant is called the common difference (d).In 4, 9, 14, 19, ..., the common difference is 5 (9 - 4 = 5). Geometric SequenceA sequence where the ratio between consecutive terms is constant. This constant is called the common ratio (r).In 3, 6, 12, 24, ..., the common ratio is 2 (6 / 3 = 2). Quadratic SequenceA sequenc...

Arithmetic Sequence Rule a_n = a_1 + (n-1)d Use this to find the nth term (a_n) of an arithmetic sequence. You need the first term (a_1) and the common difference (d). Geometric Sequence Rule a_n = a_1 * r^(n-1) Use this to find the nth term (a_n) of a geometric sequence. You need the first term (a_1) and the common ratio (r). Quadratic Sequence Rule a_n = An^2 + Bn + C This is the general form for a quadratic sequence. The coefficients A, B, and C are found by analyzing the first and second differences of the sequence.