Sequences as Functions ~adapted from Walch Education

Key Points: Sequences are ordered lists determined by functions. The domain of the function that generates a sequence is all natural numbers. A sequence is itself a function.

Key Points, continued. There are two ways sequences are generally defined—recursively and explicitly. An explicit formula is a formula used to find the nth term of a sequence. If a sequence is defined explicitly (that is, with an explicit formula), the function is given. A recursive formula is a formula used to find the next term of a sequence when the previous term or terms are known. If the sequence is defined recursively (that is, with a recursive formula), the next term is based on the term before it and the commonality between terms.

Common Difference vs. Common Ratio To determine the common difference, subtract the second term from the first term. Then subtract the third term from the second term and so on. an – an – 1 = constant To find a common ratio, divide the second term by the first term. Then, divide the third term by the second term, and so on.

Graphing Sequences Sequences can be graphed with a domain of natural numbers. Comparing the graph of a sequence, an = n – 1, to the graph of the line f(x) = x – 1 the sequence only has values where n = 1, 2, 3, 4, etc. the labels on the axes of the graph of the sequence is in terms of n and an, while the line is in terms of x and y.

Graphs Sequence Graph Line graph

Let’s Practice… Find the missing terms in the sequence using recursion. A = {8, 13, 18, 23, a5, a6, a7}

Is there a Common Difference or a Common Ratio? a2 – a1 = 5 a3 – a2 = 5 a4 – a3 = 5 The terms are separated by a common difference of 5. From this, we can deduce an = an – 1 + 5.

Using the Recursive Formula an = an – 1 + 5 a5 = a4 + 5 = 23 + 5 = 28 a6 = a5 + 5 = 28 + 5 = 33 a7 = a6 + 5 = 33 + 5 = 38 The missing terms are 28, 33, and 38.

Thanks for watching!!! ~Dr. Dambreville