Counting patterns - up to 1,000

Learning Objectives Identify if a numerical pattern is an arithmetic or quadratic sequence. Determine the rule (n-th term) for an arithmetic sequence using the formula a_n = a_1 + (n-1)d. Determine the rule (n-th term) for a quadratic sequence by analyzing first and second differences. Use the n-th term rule to find the value of any term in a sequence up to 1,000. Determine if a specific number (up to 1,000) is a term in a given arithmetic sequence. Model real-world scenarios using linear and quadratic counting patterns. Have you ever noticed the patterns in a pinecone or the way a virus spreads? 🌲 These are counting patterns that can be described with math! This tutorial moves beyond simple skip counting to explore the powerful algebraic rules that govern number patterns....

TermDefinitionExample SequenceAn ordered list of numbers, called terms, that follow a specific rule or pattern.3, 7, 11, 15, ... is a sequence where each term is 4 more than the previous one. Arithmetic SequenceA sequence where the difference between consecutive terms is constant. This constant is called the common difference (d).The sequence 10, 8, 6, 4, ... is an arithmetic sequence with a common difference of -2. Common Difference (d)The fixed amount added to each term to get the next term in an arithmetic sequence. It can be positive or negative.In the sequence 5, 12, 19, 26, ..., the common difference is d = 7. Quadratic SequenceA sequence where the second difference between consecutive terms is constant. The general form of its rule is a_n = An^2 + Bn + C.The sequence 2, 5, 10, 17,...

n-th Term of an Arithmetic Sequence a_n = a_1 + (n-1)d Use this formula to find any term (a_n) in an arithmetic sequence. 'a_1' is the first term, 'n' is the term's position, and 'd' is the common difference. Finding the Common Difference d = a_n - a_{n-1} To find the common difference (d) in an arithmetic sequence, subtract any term from the term that immediately follows it. n-th Term of a Quadratic Sequence 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, where A = (second difference) / 2.