A Sequence Is a Function
F.IF.3
Read the standard text
F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n + 1) = f(n) + f(n − 1) for n ≥ 1.
Section 1 · LearnLearn the idea
In this lesson you will: Read a_n as the nth term — the output of a function whose input is the position n; Generate terms from an explicit rule and from a recursive rule; Explain why the graph of a sequence is a set of separate points.
A list that knows its own addresses
Take the list 5, 9, 13, 17, 21. The order is not decoration. The 13 is the third entry and nothing else, and if you shuffled the list you would have a different sequence.
So every position gets exactly one value. Position 1 gives 5; position 2 gives 9; position 3 gives 13. One input, one output — that is the definition of a function from Chapter 7, and it is why sequences are filed under Functions rather than under lists.
The domain is a set of integers
Standard F.IF.3 says a sequence is a function whose domain is a subset of the integers. In practice the domain is usually 1,2,3,..., and sometimes 0,1,2,... when a problem has a natural starting amount.
There is no term number 2.5. There is no term number -4 in a list that starts at position 1. The rule may be willing to accept those inputs, but the sequence is not defined there, so they are not part of the domain. Whenever you write a sequence, say where n starts.
Two ways to hand someone the rule
An explicit rule computes any term straight from its position. A recursive rule tells you the first term and then how each term is built from the one before it.
Both descriptions below produce 5,9,13,17,21,.... The explicit one answers a_(40) in a single substitution. The recursive one matches how the situation actually unfolds — one step at a time — but reaching a_(40) means computing thirty-nine terms first.
Common mistake — Joining the dots
A student graphs a_n=4n+1, plots (1,5), (2,9), (3,13), then draws a neat line through them and reads off a value at n=2.5.
The instinct is reasonable. Every graph in Chapter 8 was a line, the points obviously line up, and a ruler makes the picture look finished.
But the line asserts that the sequence has a value at 2.5, and there is no such term. There is a second term and a third term and nothing in between. The line y=4x+1 is a different function — one that happens to pass through the sequence's points.
The same error in words sounds like this: \"the domain is all real numbers.\" It is not. It is 1,2,3,...
How to catch it: Point at a spot on your graph between two dots and ask out loud what term number it is. If the answer is \"term two and a half,\" erase the line.
Section 2 · DoPractice this standard
These published practice sets are scored on the student site and cover F.IF.3 directly.