Unit 3: Linear and Exponential Functions Coordinate Algebra Sequences

Arithmetic sequence A sequence in which the difference between consecutive terms is constant. A sequence such as 1, 5, 9, 13, 17, 21 or 12, 7, 2, -3, -8, -13 which has a constant difference between terms.

Recursive formula Pertaining to or using a rule or procedure that can be applied repeatedly. It builds onto the previous term. 2, 5, 8, 11, 14… an = an – 1 + d an = an – 1 + 3

Explicit formula An explicit formula expresses the nth term of a sequence in terms of n. It can be used to find any term in the sequence based on the term number. Let an = 2n + 5 for positive integers n. If n = 7, then a7 = 2(7) + 5 = 19.

arithmetic sequence an = a1 + (n – 1)d The first term is a1, the common difference is d, and the number of terms is n.

arithmetic sequence Example: 3, 7, 11, 15, 19 a1 = 3, d = 4, n = 5 The explicit formula is an = 3 + (n – 1)·4 = 4n – 1

Identifying an arithmetic sequence To decide whether a sequence is arithmetic, find the differences of consecutive terms. If the difference is always the same, the sequence is arithmetic. If the differences are not constant, the sequence is not arithmetic.

Writing a rule for the nth term Identify the 1st term or a1. Find the common difference. Write the general rule using an = a1 + (n – 1)d inserting the values of a1 and d. Simplify

Finding the nth term given a term and the common difference Begin by finding the first term using an = a1 + (n – 1)d and substituting for an, n and d. Write the general rule using inserting the values of a1 and d. Simplify

geometric sequence an exponential function that results in a sequence of numbers separated by a constant ratio constant ratio: the number multiplied by each consecutive term in a geometric sequence