Unit 8: Sequences and Inequalities



Unit 8: SequencesLearning Target:We will…Identify a sequence as arithmetic or geometricDetermine the way a sequence is changingFind the value of any term in the sequenceSequencesIn 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: ArithmeticPattern: 4 – 6 = -2Algebraic expression: n + (-2) or n – 2Example 2:81, 27, 9, 3, 1, 1/3, …Type of Sequence: GeometricPattern: 27÷81 = 1/3Algebraic Expression: 1/3n or n÷3Directions: Using the reading above, answer the questions below:What is a mathematical sequence?What is an arithmetic sequence?What is a geometric sequence?What kind of sequence is this? How do you know?3, 6, 12, 24, 48, …What is the pattern in the sequence above? Can you write it as an algebraic expression?What kind of sequence is this? How do you know?2, 6, 10, 14, 18, 22, …What is the pattern in the sequence above? Can you write it as an algebraic expression?Unit 9: InequalitiesLearning Target:We will:Solve one and two step inequalitiesGraph the solutions of inequalitiesIdentify inequalitiesInequalitiesIn 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:DistributeCombine Like TermsUse SADMEP to solveInequalities 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 -2x < -5Since 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:Graph the point on a number line that x is greater than (or equal to), or less than (or equal to)If there is a > or <, leave the point as an open circle ○If there is a ≥ or ≤, shade in the circle ●Choose the number to the right of the point (i.e. if 4 choose 5). Plug it in for x.If the inequality plugged in is true, draw a line on the right side of the point.If the inequality plugged is false, draw a line on the left side of the pointExample 1:Example 2:Directions: Using the reading above, answer the questions below.What is an inequality? What four symbols show you that something is an inequality?What is the only difference in solving inequalities versus solving equations?x > 6, would this inequality have an open circle or a closed circle?x ≥ -5, would this inequality have an open circle or a closed circle? ................
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