Sequences and Their Rules

In Patterns That Count, three groups described the same growing pattern three different ways — “start at 3 and double”, “term $n$ is $3\cdot 2^{n-1}$”, and a table — and a good argument broke out about whether they had found the same rule. They had. A sequence is an ordered list of numbers, and its rule can be written in several costumes, each revealing something the others hide.

A function that walks in steps

A sequence is secretly a function whose domain is the natural numbers: term 1, term 2, term 3, with nothing in between. That makes it a discrete function — graph one and you get equally spaced dots, not a connected curve. Compare $f(x)=2x$ on all real numbers (a solid line) with $f(n)=2n$ on the naturals (a string of dots climbing the same slope). Same rule, different domain, different object — a distinction Domain and Range taught you to respect, and one that matters when a model counts things that only come whole.

Three ways to write the rule

For the sequence $3,6,12,24,\dots$:

Representation	Written as	What it shows best
Recursion formula	$t_1=3$, $t_n=2t_{n-1}$	how each term grows from the last
General term	$t_n=3\cdot 2^{n-1}$	any term directly — no climbing
Function notation	$f(n)=3\cdot 2^{n-1}$	it is a discrete function

The recursion is how patterns feel — “double the last one” — but it makes you climb through every rung to reach term 40. The general term teleports straight there. Fluency means translating freely: given any one representation, produce the others.

Arithmetic, geometric, or neither

Two families dominate this unit. An arithmetic sequence adds a common difference $d$ each step, and its general term is

$$t_n=a+(n-1)d$$

— a discrete cousin of the linear function. A geometric sequence multiplies by a common ratio $r$ each step:

$$t_n=a{r}^{n-1}$$

— a discrete cousin of The Exponential Function. To classify, interrogate consecutive terms: equal gaps mean arithmetic, equal ratios mean geometric, and plenty of good sequences — $1,4,9,16$, or the Fibonacci numbers — are honestly neither. The photographs in Visual Patterns keep this classifying eye sharp all semester.

Series asks the natural next question — what do the terms add to? — and Sequences, Series, and Interest Practice covers this whole arc.

Curriculum connection

C1.1

make connections between sequences and discrete functions, represent sequences using function notation, and distinguish between a discrete function and a continuous function [e.g., $f(x)=2x$, where the domain is the set of natural numbers, is a discrete linear function and its graph is a set of equally spaced points; $f(x)=2x$, where the domain is the set of real numbers, is a continuous linear function and its graph is a straight line]

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C1.4

represent a sequence algebraically using a recursion formula, function notation, or the formula for the nth term [e.g., represent 2, 4, 8, 16, 32, 64, … as $t_1=2$; $t_n=2t_{n-1}$, as $f(n)=2^n$, or as $t_n=2^n$, or represent $\frac{1}{2}$, $\frac{2}{3}$, $\frac{3}{4}$, $\frac{4}{5}$, $\frac{5}{6}$, $\frac{6}{7}$, … as $t_1=\frac{1}{2}$; $t_n=t_{n-1}+\frac{1}{n(n+1)}$, as $f(n)=\frac{n}{n+1}$, or as $t_n=\frac{n}{n+1}$, where $n$ is a natural number], and describe the information that can be obtained by inspecting each representation (e.g., function notation or the formula for the nth term may show the type of function; a recursion formula shows the relationship between terms) Sample problem: Represent the sequence 0, 3, 8, 15, 24, 35, … using a recursion formula, function notation, and the formula for the nth term. Explain why this sequence can be described as a discrete quadratic function. Explore how to identify a sequence as a discrete quadratic function by inspecting the recursion formula.

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C2.1

identify sequences as arithmetic, geometric, or neither, given a numeric or algebraic representation

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C2.2

determine the formula for the general term of an arithmetic sequence [i.e., $t_n=a+(n-1)d$ ] or geometric sequence (i.e., $t_n=a{r}^{n-1}$), through investigation using a variety of tools (e.g., linking cubes, algebra tiles, diagrams, calculators) and strategies (e.g., patterning; connecting the steps in a numerical example to the steps in the algebraic development), and apply the formula to calculate any term in a sequence

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