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Secondly, the ratios 0 and 1 are excluded. However, r can be negative; in that case the terms are alternatively positive and negative. Example 3 The first two terms of a geometric sequence are 10 and Show that the first term greater than is the fiftieth. Each term of the sequence is greater than the preceding term.
You therefore have to show that the 49th term is less than , and that the 50th term is greater than So the first term greater than is the fiftieth. Example 4 Show that there are two geometric sequences whose first term is 5 and whose fifth term is For each of these sequences, find the tenth term. The two sequences are 5, 10, 20, 40, 80, ….
Homework C Homework Questions — Geometric Sequences 1. For each of the following geometric sequences find the common ratio and the next two terms. Find the number of terms in each of these geometric progressions. Different numbers x, y and z are the first three terms of a geometric progression with common ratio r, and also the first, second and fourth terms of an arithmetic progression.
Different numbers x, y and z are the first three terms of a geometric progression with common ratio r, and also the first, second and fifth terms of an arithmetic progression.
Related Rates Course: Lesson Objective s: Solve problems in which there are two quantities whose rates of change are related. Learn new terms in related rates problems. Give the general strategy in solving related rates problems.
Recap the lesson and talk briefly the content of the next. Connected Rates of Change: Use of the Chain Rule Consider one or two following introductory examples. Introductory example 1 Police are 30 feet from the side of the road.
Their radar sees your car approaching at 80 feet per second when your car is 50 feet away from the radar gun. The speed limit is 65 miles per hour which translates to 95 feet per second. Are you speeding?
First, draw a diagram of the setup. Let D be the distance between the police and the car and x be the distance from the car to the foot of the altitude from the police to the road. In this kind of problems, it is important to follow the following steps: Step 1: We then use the chain rule to link what we are told with what we want: Now, plug in the instantaneous numerical values: As the cone forms, the height, h cm, remains equal to the base radius, r cm, of the heap.
Give your answer to 2 significant figures. Find the rate at which the area is increasing at the instant when the radius is 50cm. You are told … You want to find … remember to look at the units for clues.
Step 2: The linking equation, using the chain rule is: In part a we had an expression for. Find the rate at which the radius is decreasing at the instant when the radius of the balloon is 2cm.
Give your answer correct to 2 significant figures. The formula relating to volumes of spheres is: Permutations and Combinations Course: Mathematics Date: Permutations 6. Combinations 6. Examples 6. Exercises Academic Standards: Standard 1 — Counting Techniques Students develop an understanding of combinatorial reasoning, using various types of diagrams and the fundamental counting principle to find numbers of outcomes and related probabilities.
They also use simulations to solve counting and probability problems. Performance Objectives: Permutations and combinations will also be on a paper and pencil test that will be assessed at the end of the chapter.
Students should also recognize when to use permutations or combinations instead of the other counting techniques when solving a problem. This assignment will be due at the beginning of the next class. The goal of the daily assessment is to give extra practice and test the knowledge and understanding of the lesson of that day. In the homework assignment the students will be applying permutations and combinations to solve various problems.
This will help give the teacher feedback to where the students are at. There middle of the chapter to help test the students overall knowledge of the lessons. Advanced Preparation by Teacher: The teacher should be prepared to teach the lesson and have all the accommodation ready for the gifted and talented students. The teacher should have the agenda, homework assignment, and bell ringer ready when students come to class.
The teacher should already have the groups divided out on paper for when the teacher chooses groups. The teacher should have all the examples ready when students come to class, with appropriate solutions. Essential Questions: Some questions on the SAT begin: Use the Fundamental Counting Permutation, Combination, principle permutations to count counting principle Procedure: Bell Ringer: A bowl contains ten red balls and ten blue balls. A woman selects balls at random without looking at them. How many balls must she select to be sure of having at least three balls of the same color?
How many balls must she select to be sure of having at least three blue balls? It is good to provide a little review before you start a new lesson.
The agenda for the day should also be written on the board for the students when they come in. While the students are working on the bell ringer activity the teacher should use the time to take attendance and make any further preparations for the lesson.
When the students have finished the bell ringer activity it is important for the teacher to go over that activity. Make sure that all of student are on the same page before you begin the new lesson.
The teacher should begin with an example on the board. The teacher knows that the answer is way too large for the students to be able to write down all the possibilities, but the teacher should let the students try anyway for about 2 to 3 minutes. The teacher should ask the students what the highest number they came up with was, and then the teacher should tell them the answer is Step by Step Procedure: Then teacher should explain exactly what each one of those words means.
Activity — have the students put in the people in the first and 2nd place race and see how many possibilities that they get. It is best to start with the question, how many ways can I award 1st and 2nd place to 4 people? The reaction that students may initially have is 4! Or So, let them place the people and record the answers.
They should get 12possibilities. Ask the students to use the same pieces, but answer, what if there was 1st, 2nd, and 3rd place to award for those 4 people? Now there are 24 possibilities. What if there is only 1st place? Now there are only 4 possibilities. When you take sub-groups from an original group the possibilities change depending on the size of the sub-group.
This does use the Fundamental Counting principle, but it extends it to something called a permutation. The formula for Permutation is defined to be when we n! If there are 4 people and you need to pick a 1st, and 2nd place winner, how many ways that be done?
Start with asking the question, how many ways can we take four people and put them in teams of 2? They many do a permutation and answer Have them work the people and record the possibilities to see. With those same people, have them calculate a team of 1 and then a team of 3?
Have them compare their results to those when we did 1 , 2 , and 3rd st nd place. Why is this different? Combinations n is the total group and r is the number in the sub- n! The other way it is written is Cnr or C n,r. See if students can find it on their calculator. Focus on canceling out the factorials and doing order of operations properly. A permutation is a set of distinct objects is am ordered arrangement of these objects. A combination is a set of unordered elements from the total set.
The teacher should ask the students if they see the difference between the two definitions. The teacher should ask the student what they think it means, and how it affects the problem. The teacher should then give a simple example of a permutation problem followed by a simple example of combination problem no need to solve yet.
The teacher should then explain the difference between the two problems and why the first is a permutation and the second is a combination. Once the teacher feels that the class understands the difference between the two, then it is time to move on to formulas.
The formula for permutation is n! The teacher should then go over some examples and teach the students how to apply the formula. Then the teacher should have the students do some examples by themselves at their desk.
Example 2 Suppose that there are eights runners in a race. The winner receives a gold medal the second-place finisher receives a silver medal, and the third-place finisher receives a bronze medal. How many different ways are there to award these medals, if all possible outcomes of the race can occur and there are no ties? Example 3 Suppose that a saleswoman has to visit eight different cities. She must begin her trip in a specified city, but she can visit the other seven cities in any order she wishes.
How many possible orders can the saleswoman use when visiting the cities? The first city is locked in but the order seven can be chosen however she wants them to be. Thus the answer is 7! Since we have a block of 3 that have to occur, the other six letters can be in any order, thus the answer is 6! Now the teacher should do the same with combinations.
The teacher should explain the combinations formula, which is: The teacher should then go over examples as a class then have the students do one example by themselves. Example 5 How many ways are there to select five players from a member tennis team to make a trip to a match at another school? How many ways are there to select a first prize winner, second-prize winner, and a third-prize winner from five different people who have entered a contest?
Example 7 A group of 30 people have been trained as astronauts to go on the first mission to Mars. How many ways are there to select a crew of six people to go on this mission assuming that all crew members have the same job?
The positions of r 1s in a bit string of length n form an r-combination of the set 1,2,3, Hence, there are C n,r bit strings of length n that contain exactly r 1s.
Now the teacher should divide the class into six groups of 3 or 4. First position the men and then consider possible positions for the women.
The teacher should then go over those problems with the class as a whole and answer in questions they might have. Then the teacher should have the each group go to the board and teach the class how to solve their problem. The group will first have to decide which counting technique to use to solve their given problem. Examples 10 1 How many license plates can be made using either three digits followed by three letters or three letters followed by three digits? When everyone is done, then one by one each group should teach the class how to solve their problem and why they used the counting principle that they did.
The teacher should close by talking real briefly about each one of the counting techniques, and explain when to use the correct ones. The teacher should then have the student restate the formulas that the lesson covered today. Then the students should go back to their seats and make sure everything is ready for the next class.
The students should begin their homework until the bell rings.
The most typical restrictions are: Method 1: First, consider the consonants: Next, consider the vowels. Since they must be together, they will occupy exactly one slot in between the consonants or at the two ends: There are ways of arranging these among themselves in the slot. Now put it all together. Method 2: A commonly taught way is to think of the 4 vowels as a bundled unit. The consonants are each of unit.
First, there are 7 units altogether, and 7! Ways to arrange these units 6 consonants and 1 bundled unit. Next, the number of permutations to arrange vowels within their unit 4! The complementary case to i includes those cases where only 2 or 3 vowels are together. However, what we want here is none of the vowels are adjacent. Recall from i Method 1 First, consider the consonants: As they must be separate, they will occupy exactly four slots in between the consonants or at the two ends: The vowels are: There are ways of arranging these among themselves in the 2!
Side-by-side comparison between i and ii: As Next, consider the vowels.