Lesson 5

Describe how to produce one step of the pattern from the previous step.

 

Here is a visual pattern of dots. The number of dots $D(n)$ is a function of the step number $n$.

1. What values make sense for $n$ in this situation? What values don't make sense for $n$?
2. Complete the table for Steps 1 to 5.

   $n$	$D(n)$
   1	1
   2	$D(1)+2=3$
   3	$D(2)+3=6$
   4
   5

3. Following the pattern in the table, write an equation for $D(n)$ in terms of the previous step. Be prepared to explain your reasoning.

Use the first 5 terms of each sequence to state if the sequence is arithmetic, geometric, or neither. Next, define the sequence recursively using function notation.

1. $A$: 30, 40, 50, 60, 70, . . .
2. $B$: 80, 40, 20, 10, 5, 2.5, . . .
3. $C$: 1, 2, 4, 8, 16, 32, . . .
4. $D$: $1, \frac12, \frac14, \frac18, \frac{1}{16},$ . . .
5. $E$: 20, 13, 6, -1, -8, . . .
6. $F$: 1, 3, 7, 15, 31, . . .

Sometimes we can define a sequence recursively. That is, we can describe how to calculate the next term in a sequence if we know the previous term.

Here’s a sequence: 6, 10, 14, 18, 22, . . . This is an arithmetic sequence, where each term is 4 more than the previous term. Since sequences are functions, let's call this sequence $f$ and then we can use function notation to write $f(n) = f(n-1) + 4$. Here, $f(n)$ is the term, $f(n-1)$ is the previous term, and + 4 represents the rate of change since $f$ is an arithmetic sequence.

When we define a function recursively, we also must say what the first term is. Without that, there would be no way of knowing if the sequence defined by $f(n) = f(n-1) + 4$ started with 6 or 81 or any other number. Here, one possible initial condition is $f(1) = 6$. (It could also make sense to number the terms starting with 0, using $f(0) = 6,$ and we'll talk more about this later.)

Combining this information gives the recursive definition: $f(1) = 6$ and $f(n) = f(n-1) + 4$ for $n \ge2$, where $n$ is an integer. We include the $n\ge2$ at the end since the value of $f$ at 1 is already given and the other terms in the sequence are generated by inputting integers larger than 1 into the definition.