Arithmetic Sequences ~adapted from Walch Education
Arithmetic sequences An arithmetic sequence is a list of terms separated by a common difference, the number added to each consecutive term in an arithmetic sequence. An arithmetic sequence is a linear function with a domain of positive consecutive integers in which the difference between any two consecutive terms is equal. The rule for an arithmetic sequence can be expressed either explicitly or recursively.
Arithmetic sequences, continued. The explicit rule for an arithmetic sequence is a n = a 1 + (n – 1)d, where a 1 is the first term in the sequence, n is the term, d is the common difference, and a n is the nth term in the sequence. The recursive rule for an arithmetic sequence is a n = a n – 1 + d, where a n is the nth term in the sequence, a n – 1 is the previous term, and d is the common difference.
Practice Write a linear function that corresponds to the following arithmetic sequence. ▫8, 1, –6, –13, …
Solve the Problem: Find the common difference by subtracting two successive terms. 1 – 8 = –7 Identify the first term (a1). a 1 = 8 Write the explicit formula. a n = a 1 + (n – 1)d a n = 8 + (n – 1)(–7)
Write the formula in function notation. Simplify the explicit formula a n = 8 – 7n + 7 a n = –7n + 15 ƒ(x) = –7x + 15 Note: the domain of an arithmetic sequence is positive consecutive integers.