Section 9 - 1: Mathematical Patterns

Section 9 - 1: Mathematical Patterns

GOAL: To be able to identify patterns between numbers.

Multiple Choice

For the function

 $f\left(x\right)=3x+9$  find f(1), f(2), f(3), f(4).

• 17, 20, 23, 26

• 9, 12, 15, 18

• 12, 15, 18, 21

• 13, 14, 15, 16

Multiple Choice

For the function

 $f\left(x\right)=2\cdot3^x$  find f(1), f(2), f(3) & f(4)

• 6, 36, 216, 1296

• 6, 12, 24, 36

• 6, 12, 18, 24

• 6, 18, 54, 162

Poll

The set of numbers {6, 18, 54, 162,...} is known as a sequence. Where we say that 6 is the first term of the sequence. A sequence usually follows a pattern beyond the first term; what is the pattern that this sequence is following?

• Start at 6 and add 12 to find each subsequent term.

• Start at 6 and multiply by 3 to find each subsequent term.

Multiple Choice

Identify a pattern and then find the next term given the sequence; {-48, -24, -12, -6}

• divide by 2; -3

• subtract 6; 0

• subtract 24; 2

• divide by 2; -2

Multiple Choice

Identify the pattern and then find the next term in the pattern; {34, 28, 22, 16, ___}

• subtract 6; -9

• subtract 18; -2

• multiply by

  $\frac{14}{17}$ ; $\frac{94}{7}$

• subtract 6; 10

Multiple Choice

Identify a pattern and find the next term in the squence; {2, -3, 4, -1, __}

• alternate subtract 5 and subtract 7; -8

• alternate subtract 5 and add 7; 6

• alternate multiply by $-\frac{3}{2}$ and $-\frac{1}{4}$ ; $\frac{1}{4}$

• alternate multiply by $-\frac{3}{2}$ and multiply by $-\frac{4}{3}$ ; $\frac{4}{3}$

Open Ended

In the figure, how does the number on the CENTER tile relate to the numbers on the other tiles?

The Rule

The explicit formula is the rule that defines the sequence. It is explicit because you can simply PLUG IN the term number, n, you are trying to find.

Multiple Choice

Which of the following would NOT be one of the first 7 terms of the sequence?

• 1

• 3

• 4

• 7

• 10

Recursive Definition

You can look at the first few terms of the sequence generated by a "rule" to see if it can be defined in another way. For example, we can say that this sequence is simply adding 3 to the previous number when you DEFINE where you start, 1.

The Recursive Definition

Let's consider the pyramid growth at right. We start with 1 block and then need to add two to add a level and then three and then four. How many blocks in each pyramid? How many blocks would be in the 12th pyramid if we continued with this same pattern?

Writing the definition

Need to consider the pattern...

Open Ended

What is the relationship between the TERM NUMBER and how many blocks you are adding?

Multiple Choice

Which of the following would be the recursive definition for this example?

• $a_n=a_{\left(n-1\right)}+3$

• $a_{n\ }=a_{\left(n-1\right)}+a_n$

• $a_n=a_{\left(n-1\right)}+n$

Multiple Choice

Find the first five terms;

Multiple Choice

What is a recursive definition for the sequence {0, -3, -6, -9, ...}

Multiple Choice

What is the explicit formula for the sequence {0, -3, -6, -9, ...}?

• $a_{n\ }=\ n\ -\ 3$

• $a_{n_{\ }=\ }-3n\ +\ 3$

• $a_{n\ }=\ \frac{1}{3}n-3$

• $a_{n\ }=-\ 3n$

Multiple Choice

Looking ahead: Over the last 40 years, the population of a city has increased by roughly 2% each year. If the population was 220,000 at the beginning of this period (a₀) what was the population 10 years later?

• about 224,490

• about 44,000

• about 268,179

• about 264,000