THE MATH FILES
Unit 8: Sequences
Learning Target:
We will…
1. Identify a sequence as arithmetic or geometric
2. Determine the way a sequence is changing
3. Find the value of any term in the sequence
Sequences
In mathematics, a sequence is a list of numbers in a specific order with a pattern. The numbers in a
sequence are called terms, and our goal as mathematicians is to determine how the sequence is
changing from term to term. Our goal is to determine the change from one number to the next. Look
at the sequence below:
2, 5, 8, 11, 14, 17,…
You probably notice some pattern in the sequence. Do you see it? To get from one term to the next
you must always add 3. You can find the number we are adding by choosing a term and then
subtracting the term that came before it (i.e. 5 – 2 = 3). We can then check by using the next term and
making sure it still works (i.e. 8 – 5 = 3).
Sometimes we like to write the sequence as an algebraic expression where a variable (n) takes the
place of a term in the sequence and we write down how to get from one term to the next. In this case,
we could write that the sequence from one term to the next is n + 3, because we must add three to
each term to find the next term. Whenever we are adding or subtracting to get from one term to
another in a sequence it is called an arithmetic sequence.
Sometimes, however, we have a sequence that is a little more complicated, like the one below.
4, 8, 16, 32, 64,…
In the event that we have a sequence like this, there is no single number that we can add to each
term to find the next one (i.e. 8 – 4 = 4 but 16 – 8 = 8, so 4 does not work for every term). Instead we
are multiplying each term by the same number to find the next number in the sequence. You can find
the number we are multiplying by dividing one term by the term that came before it in the sequence
(i.e. 8 ÷ 4 = 2), and then checking (16 ÷ 8 = 2). Therefore to get from one term to the next in the
sequence you must multiply that term times 2. This would mean that our variable expression is 2n.
We call sequences that are multiplied to get from one term to the next geometric sequences.
Example 1:
6, 4, 2, 0, -2, …
Type of Sequence:
Pattern:
Algebraic expression:
Arithmetic
4 – 6 = -2
n + (-2) or n – 2
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Example 2:
81, 27, 9, 3, 1, 1/3, …
Type of Sequence:
Pattern:
Algebraic Expression:
Geometric
27÷81 = 1/3
1/3n or n÷3
Directions: Using the reading above, answer the questions below:
1. What is a mathematical sequence?
2. What is an arithmetic sequence?
3. What is a geometric sequence?
4. What kind of sequence is this? How do you know?
3, 6, 12, 24, 48, …
5. What is the pattern in the sequence above? Can you write it as an algebraic expression?
6. What kind of sequence is this? How do you know?
2, 6, 10, 14, 18, 22, …
7. What is the pattern in the sequence above? Can you write it as an algebraic expression?
THE MATH FILES
Unit 9: Inequalities
Learning Target:
We will:
1. Solve one and two step inequalities
2. Graph the solutions of inequalities
3. Identify inequalities
Inequalities
In mathematics, inequalities represent two expressions or an expression and a term (a term is just
another name for a number) that are not equal to each other. They always have these signs: <,>,≤,≥.
These mean that one expression is not equal to the corresponding term or expression.
Solving inequalities is just like solving equations. You must:
1. Distribute
2. Combine Like Terms
3. Use SADMEP to solve
Inequalities only have one major exception: IF YOU MULTIPLY OR DIVIDE THE VARIABLE BY A
NEGATIVE NUMBER, YOU MUST FLIP THE INEQUALITY.
Please note the example below.
-2x – 3 > 7
+ 3 +3
-2x > 10
-2
-2
x < -5
Since I am dividing the variable by a negative number, I must therefore flip the inequality.
Inequalities are also important, because you can graph the solution to linear inequalities. (A linear
inequality just means that it is an inequality for which we are solving for x). In order to graph the
solution to linear inequalities we must determine what the inequality is telling us above the variable.
For instance, x < 6 is telling me that our variable could equal any number less than 6, while x ≥ -2,
tells me that x must be either -2 or any number greater than -2. We are graphing all of the solutions to
this inequality. If x < 6 then any number less than 6 is a solution, and we need to show that.
In graphing inequalities, you must complete three steps:
1. Graph the point on a number line that x is greater than (or equal to), or less than (or equal to)
2. If there is a > or <, leave the point as an open circle ○
If there is a ≥ or ≤, shade in the circle ●
3. Choose the number to the right of the point (i.e. if 4 choose 5). Plug it in for x.
a. If the inequality plugged in is true, draw a line on the right side of the point.
b. If the inequality plugged is false, draw a line on the left side of the point
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Example 1:
Example 2:
Directions: Using the reading above, answer the questions below.
1. What is an inequality? What four symbols show you that something is an inequality?
2. What is the only difference in solving inequalities versus solving equations?
3. x > 6, would this inequality have an open circle or a closed circle?
4. x ≥ -5, would this inequality have an open circle or a closed circle?