Number sequences

Learning Objectives Identify and describe arithmetic, geometric, and simple quadratic number sequences. Determine the common difference or common ratio of a sequence. Find the next several terms in a given number sequence by identifying its pattern. Derive and use the formula for the n-th term of an arithmetic sequence. Derive and use the formula for the n-th term of a geometric sequence. Find the formula for the n-th term of a quadratic sequence using the method of finite differences. Apply sequence formulas to solve for a specific term in a sequence. Ever notice how a pyramid is built layer by layer, or how a virus can spread exponentially? 🦠 These are real-world examples of number sequences! In this tutorial, we will explore the fascinating world of number patterns, k...

TermDefinitionExample SequenceAn ordered list of numbers, called terms, that follow a specific rule or pattern.The list 3, 6, 9, 12, ... is a sequence where the rule is 'add 3 to the previous term'. Term (a_n)Each individual number in a sequence. The notation `a_n` refers to the term in the n-th position.In the sequence 10, 20, 30, 40, ..., the 3rd term, or `a_3`, is 30. Arithmetic SequenceA sequence where the difference between any two consecutive terms is constant. This constant is called the common difference (d).2, 7, 12, 17, 22, ... is an arithmetic sequence with a common difference of 5. Geometric SequenceA sequence where the ratio between any two consecutive terms is constant. This constant is called the common ratio (r).3, 6, 12, 24, 48, ... is a geometric sequence with...

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. N-th Term of a Geometric Sequence a_n = a_1 * r^(n-1) Use this formula to find any term (`a_n`) in a geometric sequence. `a_1` is the first term, `n` is the term's position, and `r` is the common ratio. N-th Term of a Quadratic Sequence a_n = An^2 + Bn + C The formula for a quadratic sequence. The coefficient `A` is half of the constant second difference. `B` and `C` can be found by solving a system of equations or by analyzing the linear sequence that remains after subtracting `An^2` from the original sequence.