Distillation in Chemistry - What You Should Know

Distillation in Chemistry - What You Should KnowDistillation in Chemistry is a process that involves the separation of components that have been taken out of a liquid solution by cooling down the solution and then heating it. It was a well known process used in the olden days of chemistry but is no longer common nowadays.During this process, the components will be separated from the solution and will undergo a chemical reaction in order to be mixed with other components. The reaction is known as vaporization and the components are classified as hydrophilic or hydrophobic depending on the surface they will be on. Hydrophilic components will dissolve in water, while the hydrophobic components will not.This process helps in chemical separation. You can find many uses for this process but there are a lot of different types of distillation. Below are some of the methods you can find.Boiling: this process has been in use since ancient times when people used to boil water to test its qualit y. It was also used to create lye. It is considered to be one of the best methods to test the purity of the liquids because of its simplicity. It is also considered to be one of the cheapest ways to test the quality of the solution.The chemical that is boiled will be separated from the solvent that is put into the flask during the process. The different chemicals that will be produced will be known as ethyl acetate, sulfuric acid, and so on. In the end, the solvent will evaporate and will leave the element that is boiled in the flask.The chemical extraction is the same as boiling except that the solvent will evaporate. The heat will help in separating the different components. The chemist will have to add the different components one by one so that the most important element remains for the chemical reaction. The physical solvents are the most effective way to distill because they have no smell and they are easy to handle. You can use several kinds of solvents to make different type s of process. Different solvents will produce different qualities of the product.

Our Advice to Progress in Violin

Our Advice to Progress in Violin How to Get Better at Playing the Violin ChaptersHow to Use a Chinrest and a Shoulder RestChoosing the Right Practice Mute for Your ViolinHow to Play the Violin Left-handed“A gift without technique is nothing more than a dirty habit” [Un don sans technique n’est rien qu’une sale manie.] - George Brassens (1921-1981), Le mauvais sujet repenti.Learning to play the violin is a great idea, especially if you're really enthusiastic about music!However, there are no frets, a complicated bow technique, complicated tuning, and you’ll probably need to spend months in violin lessons just to be able to play in key and develop your playing technique.While very few children teach themselves to play musical instruments, a lot of adults do.It’s difficult to learn any bowed string instrument. That’s why in this article, we’ve come up with some advice for getting better at the violin.violin teacher near me!

Should I Go To University of Notre Dame

Should I Go To University of Notre Dame The tutors behind Varsity Tutors are not just here to teach theyre sharing their college experiences as well. Ashley graduated from the University of Notre Dame in 2012 with a Bachelors degree in Biological Science Theology. She currently tutors many subjects in Boston specializing inBiology tutoring, ACT prep tutoring,SAT prep tutoring, Essay Editing tutoring, and more. Take a look at her review of the University of Notre Dame: VT: Describe the campus setting and transportation options. How urban or safe is the campus? Are there buses or do you need a car/bike? Ashley: The University of Notre Dame is located in South Bend, Indiana, only a few miles south of Michigan and less than two hours east of Chicago. The campus is gorgeous. It is essentially self-contained, with sweeping lawns, beautiful gardens, patches of woodland, and two lakes (with lovely trails for jogging!). I quickly felt right at home on the University of Notre Dames campus, and it is quite safebesides being set apart from the surrounding neighborhoods, the campus has ample lighting and many emergency call stations. Almost all undergraduates live right on campus, and you can walk nearly anywhere in about 15 minutes. Some students ride their bikes to class, and to get off-campus, most students drive, bike, or take a shuttle bus. VT: How available are the professors, academic advisers, and teaching assistants? Ashley: Very rarely did I have difficulty getting in contact with professors or advisers at the University of Notre Dame. Particularly for first-year or especially challenging courses, the professors and teaching assistants know that students will need help, and they often hold frequent office hours or tutorial sessions. Academic support in the form of tutoring and advising is also available, with a special focus on first-year students. VT: How would you describe the dorm life rooms, dining options, location, socialization opportunities with other students? Ashley: The University of Notre Dame is well known for its unique dorm life. Almost all undergraduates choose to live on-campus for all four years. The University of Notre Dames dorm life is special because there are no sororities or fraternities. Instead, each single-sex dorm has a distinctive character and community of its own. Each dorm has its own chapel and Mass schedule, as well as a hall council that plans events (liturgical, social, service, etc.) to support dorm life. Students are fiercely loyal to their dormsthey celebrate their dorm mascot (we were the Howard Hall Ducks), play against other dorms teams in inter-hall sporting events (the inter-hall football championship is played in Notre Dame Stadium), and compete for the celebrated title of Hall of the Year. Your dorm is truly your family while at the University of Notre Dame! There are two main dining halls on campus, and both are fantastic (I highly recommend North Dining Halls pasta stir-fry and South Dining Halls make-your-own-pizza). In addition, there are a variety of cafes and smaller dining options scattered throughout the academic buildings and LaFortune Student Center. Most students have a meal plan, which gives them access to three all-you-can-eat meals in the dining hall each day, in addition to Flex Points that can be spent at eateries around campus. VT: Which majors/programs are best represented and supported? What did you study and why? Did the university do a good job supporting your particular area of study? Ashley: The University of Notre Dame has a wide variety of majors in the College of Science, College of Arts and Letters, College of Engineering, and Mendoza College of Business. All students begin in the First Year of Studies program, where they take foundational courses that set the stage for their specialization when they declare a major for sophomore year. Most students have at least one minor in addition to their major, and many students double-major. For example, I was a Biological Science major (College of Science) with a second major in Theology (College of Arts and Letters). I loved having the opportunity to learn more about my faith by taking theology classes alongside the science classes in my Biological Science major. Even better, I had many opportunities to take elective coursessuch as Greek, geology, and literatureoutside of my main focus areas. VT: How easy or difficult was it for you to meet people and make friends as a freshman? Does Greek life play a significant role in the campus social life? Ashley: The University of Notre Dame is most often described as a family, and this strong sense of community begins with an acceptance letter that says, Welcome home. First-year students are immediately welcomed into their dorm communities, and they can participate in a variety of dorm and inter-hall events specifically designed for freshmen. In addition, the University Notre Dame hosts an event during which all of the campus clubs and organizations are present so students can browse their (nearly infinite!) options and find activities that suit their interests. First-year students also quickly get to experience the great tradition of campus football, and autumn Saturdays are a near-sacred time as everyone cheers on the Fighting Irish. VT: How helpful is the Career Center and other student support services? Do many reputable companies recruit on campus? Ashley: The Career Center is a useful resource, which I especially used when I needed help developing a resume for the first time. Juniors and seniors often attend the several career fairs offered throughout the year, and the University of Notre Dame also hosts a service fair for students who would like to commit themselves to a service or religious organization for a few years after graduation. VT: How are the various study areas such as libraries, the student union, and dorm lounges? Are they over-crowded, easily available, spacious? Ashley: The University of Notre Dames main library is 13 stories high and filled with desks and comfortable chairs for studying. In addition, there are several smaller libraries around campus. Many academic buildings (e.g., the Jordan Hall of Science) have designated study spaces that can be either quiet or social, depending on the atmosphere encouraged by that department. Dorms usually have at least one study lounge on each floor, but these can be louder and more crowded than the library, especially when students are hesitant to walk too far in the winter. I always enjoyed trying out new study spaces, as well as finding interesting, out-of-the-way locations in which to read or write a paper. VT: Describe the surrounding town. What kinds of outside establishments / things to do are there that make it fun, boring, or somewhere in between? To what extent do students go to the downtown area of the city versus staying near campus? Ashley: The University of Notre Dame is bordered by suburban neighborhoods, a larger commercial road, campus apartments, and recently-developed commercial streets geared toward students (think Chipotle, campus bookstore, ice cream, trendy restaurants, etc.). Honestly, there is so much to do that most students spend the majority of their free time on campus! The neighboring city of Mishawaka has a ton of stores and restaurants. For students willing to travel, the city of Chicago is easily accessed by bus or train, and the Lake Michigan shore is not too far away, either. VT: How big or small is the student body? Were you generally pleased or displeased with the typical class sizes? Ashley: There are about 8,000 undergraduates, which was a perfect size for meneither overwhelmingly large, nor so small that it felt constraining. As a Biological Science major, I often had larger lecture-style classes in my first two years, but class sizes decreased for more specialized upper-level courses. My Theology courses in the College of Arts and Letters were generally smaller and included many seminar-style (less than 20 students) classes. I felt that these class sizes, though diverse, were very appropriate given the nature of the courses. VT: Describe one memorable experience with a professor and/or class. Perhaps one you loved the most or one you regret the most. Ashley: My most memorable courses at the University of Notre Dame were my Beginning Greek I and II classes, which I took just for fun (as a science major in a classics course, I was definitely an anomaly!). Learning an ancient language required a skill set and study strategies that were very different from my typical homework assignments (memorizing biological processes, writing lab reports, or writing theology essays). Greek was by far my hardest and most demanding course that yearand yet it was one of the most fun! My professor did an outstanding job of making his lessons engaging and yet incredibly demanding. His high expectations gave us all the motivation to prepare meticulously for each class. Even though Greek was an elective for me, I also treasured the fact that my study of Greek simultaneously informed the other classes I was takinggeology, physiology, and theology are all disciplines with a Greek-based technical vocabulary. Check out Ashleys tutoring profile. The views expressed in this article do not necessarily represent the views of Varsity Tutors.

Mastering the Art of Follow Up - Introvert Whisperer

Introvert Whisperer / Mastering the Art of Follow Up - Introvert Whisperer Mastering the Art of Follow Up “I’ll follow up with you tomorrow.”  Only that follow up never happens.  How does that make you feel?  It makes no difference what the follow up was going to be. It can be a call back from a job interview all the way to your work associate following up to tell you if they can be in a meeting.  Besides being unprofessional, a failed follow up doesn’t make you feel good; it makes you feel forgotten and unimportant.    Hopefully, those are feelings you don’t create with others.       The follow up is especially important because it is one thing that demonstrates your integrity.  Follow up is also important for several other reasons: It helps to keep business moving forward.  We can all work with some delay in our various business processes as long as we know that the person we need something from will follow up with whatever is needed from them. It keeps people accountable.  When you follow up with someone on a commitment they made, it will keep them more focused on the deliverable and staying accountable. It is about integrity and trust.  If you’ve failed to follow up on something enough times, you’ve trained those people to not trust that you will ever do what you say. Some jobs require it.  If you are in certain jobs, you have to be the one to make the phone call or drop into some one’s office to follow up.  This happens because of the role you and the other person play.  If you don’t do the follow-up, the business won’t move forward, nor will your career. It can be a tool for your job.  Follow up is an important tool to my job as a coach, just as it was when I was a manager.  Sometimes people rely on your ability to follow up to keep things moving or to help them through a process.  It’s critical to many jobs. If you happen to be a person who lacks good follow up skills, don’t worry; here are five things you can do to go from bad to great: List your follow ups.  Many people have a running ‘to-do’ list. This trick is just like that only it falls in the category of following up.  List the required follow-up, put the date it needs to be done and check it off after you’ve done it.  People will think you are a rock star. Put it on your calendar.  Most people have some type of calendar and calendar reminder system on their phone.  When you have something to follow up on, don’t try to remember it; simply put it in your calendar to ping you at the appropriate time. Use your files.  If you keep a file for a project or person, make a sticky note and put it in the file to remind you to follow up on a specific thing related to that file. This will only work if you refer to the file somewhat frequently. Don’t get sticky note crazy.  Some people will generate hundreds of sticky notes and put them all over the place to help them remember to follow up on things.  When you do this, it loses the impact of the sticky note as a reminder.  The eye gets used to the clutter of sticky notes and can now easily ignore them. Don’t rely on your memory.  Most people can only juggle a limited amount of things in their memory at any given time.  Unless you are one of those rare people who remember everything, skip trying to remember your follow ups.  It can’t be relied upon.  Instead of trying to remember to get into the habit of using one of the simple tips mentioned above. Follow up is a professional tool used in all jobs from restaurant servers to CEOs.    If you are good at it, you will be branded as a trust-worthy peer and professional with whom other people will want to work. CAREER ACCELERATION Do you know what your next career step is? Many people don’t. I want to help you accelerate your career by connecting you with your Free Instant Access to my eBook on how to construct your Career SMART Goals â€" that will help you put together your actions and keep you accountable. Get your copy now and start your action plan today: CLICK HERE!

Biology Revision Online The Best Websites to Revise Biology

Biology Revision Online The Best Websites to Revise Biology Online Resources for Revising A Level and GCSE Biology. ChaptersWhy Use Websites to Revise?Prepare for Your Exams with A Level BiologyTry a New Way of Revising Biology with Seneca LearningFind What You Need to Learn at Biology ResourcesGrow Your Biology Knowledge with Pass My ExamsStudy Biology Well with Study WisePut Your Biology Revision in Safe Hands with BBC BitesizeLearn Biology with Love BiologyCrack those Biology Examinations with S-CoolAnd Something a Little Different…Biology is a beautiful and diverse discipline. Ranging from the microscopic organisms at the level of cells and bacteria, to the environmental issues facing ecosystems and biodiversity, biology takes in all manner of life on earth. If you think about it this way, it’s really quite an incredible subject.However, incredible or not, you still have to revise for your exams. And, whilst we will insist and insist and insist that trying to enjoy your subject is the key to success in your biology course, you may not be super-interested in the joys of the subject when you are stressing over your exam.So, let’s take a little step back and take a look at some of the key resources available to help you achieve the grades that you are striving for. For those interested, we have articles on the best biology books and textbooks for revision, on the best apps and podcasts for revising biology online, and on general biology revision advice. But here, we are focusing on websites that can see your revision through to the end. MahimaBiology Teacher 5.00 (10) £30/h1st lesson free!Discover all our tutors AndyBiology Teacher 5.00 (2) £15/h1st lesson free!Discover all our tutors AlishaBiology Teacher 5.00 (10) £9/h1st lesson free!Discover all our tutors SaifBiology Teacher 5.00 (9) £22/h1st lesson free!Discover all our tutors FrankBiology Teacher 5.00 (8) £90/h1st lesson free!Discover all our tutors HeavenliBiology Teacher £9/h1st lesson free!Discover all our tutors SiobhanBiology Teacher £60/h1st lesson free!Discover all our tutors MackenzieBiology Teacher 5.00 (2) £15/h1st lesson free!Discover all our tutorsWhy Use Websites to Revise?Websites are great for revision. Not only do they rid you of the necessity of carrying around your giant biology book all day long â€" as all the information you need can be accessed from your mobile phone.But, moreover, they are usually designed with the aim in mind of presenting information in the simplest, most accessible way possible. And, unlike with books, they can change up the format â€" giving you video, audio, slideshows, interactive quizzes, and more.This is not to knock books, mind you. Books are the indispensable resource for young scientists and students. But if you want to change up the way that you study, combining the more traditional methods with online technologies can keep you learning in different ways.Here are some of the best biology revision websites â€" for both GCSE and A Level â€" available.Prepare for Your Exams with A Level BiologyAs the name suggests, A Level Biolo gy provides revision resources for those studying towards an A Level biology exam â€" whether for OCR, Edexcel, or AQA. The site is used by both teachers and students, and you’ll find heaps of cool stuff on every subject from genetics to plant physiology to molecular and cell structure.Click through to your exam board and you’ll find tailored content specifically for you. Or else, you can go straight through to past papers, mind maps, quizzes, and PowerPoints.Some of the content is only available to premium members of the site, but even those looking for free material will find what they need.Rather than books, try websites for studying biology.Try a New Way of Revising Biology with Seneca LearningSeneca Learning is a new site (currently in beta mode â€" meaning that it is not yet fully functional) that claims to double the speed at which you revise. It’s hooking students of all subjects and now boasts over 800,000 members.The content on offer is really striking, as glossy ima ges are combined with clear and precise information presented in bullet points. The information is presented like an interactive slideshow that you can pace yourself.The presentations cover A Level, GCSE, iGCSE, and International Baccalaureate courses, and diverse topics from gene mutation, protein synthesis, and cell membrane to ecosystem management and sexual reproduction. Everything you need for the biology exam will be here.Find a biology tutor now.Find What You Need to Learn at Biology ResourcesBiology Resources is an online cupboard full of incredibly helpful documents, notes, question papers, discussion prompts, videos, and ideas for experiments. It also has a load of PowerPoint presentations for biology revision too.This is mainly an option for students and teachers of GCSEs and iGCSEs â€" and it is actually designed with teachers in mind â€" but it is a helpful resource for any level of student looking to get to grips with the basics of the subject.With good, comprehensive information on all of the biology topics in the curriculum, Biology Resources allows you to study, in depth, the organs and tissues of animals, the principal elements of the study of life, heredity, genes, and plant reproduction. MahimaBiology Teacher 5.00 (10) £30/h1st lesson free!Discover all our tutors AndyBiology Teacher 5.00 (2) £15/h1st lesson free!Discover all our tutors AlishaBiology Teacher 5.00 (10) £9/h1st lesson free!Discover all our tutors SaifBiology Teacher 5.00 (9) £22/h1st lesson free!Discover all our tutors FrankBiology Teacher 5.00 (8) £90/h1st lesson free!Discover all our tutors HeavenliBiology Teacher £9/h1st lesson free!Discover all our tutors SiobhanBiology Teacher £60/h1st lesson free!Discover all our tutors MackenzieBiology Teacher 5.00 (2) £15/h1st lesson free!Discover all our tutorsGrow Your Biology Knowledge with Pass My ExamsPass My Exams is a website specialising in revision resources for GCSE science, with significant databases of notes o n physics, chemistry, and biology.All the notes come as simple text files or as animated presentations, with cute pictures and features that make the whole revision process a little more engaging.The thing about Pass My Exams is that the content is really comprehensive, with detailed information on literally every part of the course. So, if you are after clear, compelling, and creative notes that cover everything â€" from enzymes to osmosis to photosynthesis â€" this may well be the best place to come.With a computer, you have access to hundreds of different biology resources online.Study Biology Well with Study WiseThis is one for everyone. Study Wise is a website that hosts information relevant to both A Level students and those at GCSE level â€" no matter which exam board you are studying for.Study Wise works a bit differently than the other websites, as it has a variety of contributors that upload revision worksheets, notes, and questions and answers. It’s a little more like a portal to other revision resources on the internet. Whilst this means that the information is much more widely ranging, it means that the quality is not always consistent â€" so keep your eyes peeled.However, with links to past papers and mark schemes, as well as a section on general study tips, it is a handy site, for any subject you are studying.  Look up for  A Level biology tutor  now.Put Your Biology Revision in Safe Hands with BBC BitesizeRun by the largest broadcasting organisation in the world â€" the BBC â€" Bitesize is a go-to revision resource for thousands and thousands of students across the UK and has been for many years. No matter what you are studying, you will almost certainly have come across this website in the past, and it is as good as its reputation suggests.Whilst Bitesize is undergoing something of a renovation, with new content being developed right now, the resources available for GCSE students (the new content for A Levellers is forthcoming) are really w onderful.Say you want to know about cell division. You can go through and select which exam board you are studying for and find pages and pages of gorgeously designed revision materials. You’ll find here everything from a chromosome to the structure of DNA, as well as everything related to this â€" from cloning to the nature of the nucleus.Understand the world around you, by studying biology.Learn Biology with Love BiologyLearn Biology is a website dedicated to GCSE biology â€" and it is probably the best place for online biology quizzes.The website hosts over fifteen hundred multiple choice questions, divided into lots of different specific quizzes. You have ten minutes to complete each quiz and each one gives you feedback and helpful biological information on where you went wrong (if you went wrong!). And if you don’t do as well as you want to, you can go back and do the quiz again with different questions!Alongside the main quiz, there are keyword questions, in which you have to pick the correct word for the given definition.Crack those Biology Examinations with S-CoolS-Cool boasts seven million users â€" students who have revised with the website over the years. And it claims that sixty percent of its users achieved a grade above a B. It’s not a bad record, but its revision resources would be worth a look regardless.Split into the whole raft of different GCSE and A Level topics, S-Cool provides dedicated revision banks on each. If you register with the website, once you have read the revision material, you can test yourself on it, and then receive further guides on the material that you have just covered â€" to make it sink in properly.Another perk of signing up is that you can create your own revision timetable with the tasks from the website â€" so you know precisely what progress you are making.And Something a Little Different…Alongside these sites, which are designed for revision, you can push your studies to the next level by reading around th e subject â€" and there are plenty of websites that can provide you with information that will fill you in on what’s going on in the actual world of science these days.The science pages of The Guardian, The New York Times, and the BBC, just as examples, are brilliant sources of information on contemporary discoveries in science â€" in genetics, evolutionary theory, or neuroscience.Reading stories that come from outside of your textbook â€" or outside of your school syllabus â€" can make the subject just a little more interesting, something that might help you reach the end of your revision period.

Online Statistics Homework Help - Online Tutoring

Online Statistics Homework Help - Online Tutoring Statistics is the orderly collation of data, its analysis and description. Tutor Pace is an online Statistics Homework Help service for students who may suddenly find themselves unable to cope with this difficult subject. Subjects are not intrinsically difficult or easy, it is the students thinking which makes it so. If students get bad results in their attempts to solve statistical problems they call the subject difficult. The truth is that our comprehension of the subject is poor. Tutor Pace uses its unique system of Statistics Homework Help to instill confidence in the student while at the same time solving their immediate school assignments. The popularity of Tutor Pace is based on its easy availability on the internet for twenty four hours a day throughout the year. That means students have a virtual friend on whom they can rely day and night. Students need not panic because they know Statistics Homework Help is at hand at the click of a mouse button. Check the credentials The first thing a student should do is to gather information from fellow students and their parents regarding Tutor Pace. A prospective student can even search forums on the net to get feedback on the standing of this Statistics Homework Help organization. If satisfied the student should enroll with Tutor Pace through a very secure online transaction. The student will be entitled to a comprehensive video chat session with an expert from Tutor Pace on their website. Students can ask all relevant questions about the methods employed by them for Statistics Homework Help. A vast subject Statistics is a vast subject. You have to know the basics before you progress to the advanced lessons. Each student determines the level of statistical lessons that he or she requires. This is decided in the beginning with the help of an online video chat with a tutor. The tutor assesses the strengths and weaknesses of the student and explains how the homework help will proceed in the future. A convenient time is set for the student to call up everyday and get Statistics Homework Help. The areas of assistance can be in hypothesis sampling, index numbers, law of large numbers, dispersions, averages, sampling distribution and many more aspects of statistics. There is no need in online tutoring to listen to facets of the subject a student is comfortable with. The student can jump to and from various topics. Communicating clearly Students who are able to pinpoint their problem to the tutor get the best results. Tutor Pace relies on patient teachers who make students comfortable first and then spend as much time as necessary to get details about the Statistics Homework Help required by the student. Handling software Sometimes tutors discover that the student does not have a problem with the lessons in statistics. The student may have a problem in the presentation of statistics using modern software programs. Presentation of statistics using pie charts, graphs and columns is an essential part of the subject. In this case the tutors concentrate their homework help to the student in the usage of various software programs. This is the reason for the popularity of Tutor Pace. Identification of the problem is more important than a wasteful dissemination of useless information.

GMAT Quant Strategies Summing an Arithmetic Sequence

GMAT Quant Strategies Summing an Arithmetic Sequence GMAT MBA Admissions Blog One of the more difficult GMAT problem types deals with summing an arithmetic sequence. Problem 157 in the 12th edition Official Guide is an example: 157. For any positive integer n, the sum of the first n positive integers equals (n(n + 1))/2. What is the sum of all the even integers between 99 and 301? (A) 10,100 (B) 20,200 (C) 22,650 (D) 40,200 (E) 45,150 In this problem the GMAT provides us with an equation, and we can use that equation along with a bit of algebra to solve the problem. However, this might not always be the case, and no doubt in the future the test writers will introduce problems without helpful hints - indeed there are a couple in the 12th and 13th editions of the OG. Consequently, students should know how to do this type of problem in general. First a definition: Arithmetic sequencesare ordered lists of numbers (on the GMAT these numbers are most likely integers) where the difference between adjacent numbers is constant. The most common example on the GMAT are lists of consecutive integers. For example: {1, 2, 3, 4, 5, 6}, or, in general, any list of the form {x, x + 1, x + 2, x + 3...} Arithmetic sequences are similar, but the difference between the terms can be anything. For example: {3, 7, 10, 13, 16}, {1/2, 5/2, 9/2, 13/2}, {2, 4, 6, 8, 10, 12, 14}, or, in general {x, x+ d, x + 2d, x+ 3d...} Everything we need to know about an arithmetic sequence can be summed up in three pieces of information: 1. The first (or last) term 2. The difference between the terms 3. The number of terms For the most part, the first two pieces of information are explicitly or implicitly stated. If we are summing even integers, the difference is two and the first and last terms must both be even regardless of the range stated in the problem. The third piece of information is something we generally have to derive for ourselves. I'm going to use a simple example to explain how to do this. How many integers are there between 7 and 15, inclusive? Intuitively, it seems like we should find the difference between the two numbers, 15 - 7 = 8, and that will be the number of terms. If we take the brute force approach of actually listing the integers we find that the intuitive approach is off by 1: {7, 8, 9, 10, 11, 12, 13, 14, 15}. If you count the integers, you'll find that there are nine of them in the list. It might seem like we should just take the brute force approach every time we need to solve this sort of problem, but for larger sets this would be impractical. So, why is the intuitive approach incorrect? It pretty simple - by subtracting the 7, we don't include it in the set, and the count is off by one. To compensate for this, we add one to the difference. If x is less than y, the number of integers between xand y, inclusiveis y- x + 1 We have to modify this formula if we want to use it count an arithmetic sequence where the difference between terms is greater than 1. If d is the difference between the terms in the sequence, and xis less than y, then the number of terms in the sequence is (y- x)/d + 1 We have to be a bit careful applying this formula. If we're asked to find the number of multiples of 3 between 2 and 34, our first term is not 2, it's 3, and our last term isn't 34, it's 33 - this is because we're only counting multiples of 3. So, now that we know how to count the number of terms we can move on to finding the sum of the terms. First we need a couple of definitions 1. The averageor arithmetic meanof a set of numbers is The sum of the terms/The number of terms For example, the average of the first 5 positive even integers (2, 4, 6, 8, and 10) is (2 + 4 + 6 + 8 + 10)/5 = 30/5 = 6 2. The medianof a set of ordered numbers is the middle term of the set or the average of the two middle terms. That is, if an ordered set has 5 numbers, the 3rd term will be the median; if an ordered set has 6 numbers, the average of the 3rd and 4th terms will be the median. Two examples: To find the median of the set {8, 17, 7, 13, 4}, we order the terms of the set from least to greatest (or vice versa) and find the middle term. The middle term of the set {4, 7, 8, 13, 17} is 8. So the median of the set is 8. To find the median of the set {8, 17, 7, 13, 4, 31} we follow the same procedure, but now we take the average of the two middle terms. The average of the two middle terms of the set {4, 7, 8, 13, 17, 31} are 8 and 13. So the median of the set is (8 + 13)/2 = 10.5 Arithmetic sequences of numbers have the nifty property that the median and mean of the set are the same number. Mean of an arithmetic sequence = Median of an arithmetic sequence Furthermore the mean of an arithmetic sequence is just the average of the first and last term, and since the median is equal to the mean, the median is also the average of the first and last term. If this seems mysterious, remember that there is a geometric interpretation of the mean. The mean of two numbers on the number line is the midpoint, e.g. the mean of 1 and 7 is 4, because the distance from 1 to 4 is equal to the distance betwen 4 and 7. In an arithmetic sequence the midpoint between the first and last term is exactly the median. Now let's derive the formula for the sum of an arithmetic sequence. Recall that Mean = Sum of terms/Number of terms Therefore (Mean)(Number of terms) = Sum of terms So, if xis the first term of an arithmetic sequence,yis the last term, and d is the difference, the sum of the sequence is [(x+ y)/2][((y - x/)d) + 1] = Sum of terms Now let's return to problem 157 from the 12th OG: 157. For any positive integer n, the sum of the first n positive integers equals (n(n + 1))/2. What is the sum of all the even integers between 99 and 301? (A) 10,100 (B) 20,200 (C) 22,650 (D) 40,200 (E) 45,150 Because we are only concerned with the even integers, the first term of the sequence is 100 and the last term of the sequence is 300. The mean of the sequence is (100 + 300)/2 = 400/2 = 200 The number of terms in the sequence is ((300 - 100)/2) + 1 = (200/2) + 1 = 100 + 1 = 101 Thus, the sum of the sequence is (101)(200) = 20,200 The explanation is the OG is much more convoluted, with unnecessarily obscure notation. The method presented here is clean and quick. Try it for yourself. 1. Find the sum of the first 100 positive integers 2. Find the sum of all the multiples of 3 between 1 and 100 3. Find the sum of the multiples of 7 between 47 and 624 4. The sum of a set of 50 consecutive positive odd numbers is 172. Find the fist and last term of the sequence I hope you enjoyed this post. I will post answers to the questions with explanations in a subsequent post. Please visit our GMAT Tutoring page to learn more about MyGuru's approach to test prep. About the Author John E. is a Senior MyGuru GMAT tutor in Chicago. He also tutors the GRE, SAT, ACT, and almost anything math related. He holds a B.S. in Mathematics and English from Indiana University, and a M.S. in Mathematics from UIC. He's scored in the 99th percentile on all of the above standardized tests, and has worked for MyGuru and other companies to develop proprietary test prep questions and explanations of difficult quantitative concepts to help students improve their GMAT scores. He's currently working with MyGuru to develop proprietary test prep material focused on the most difficult GMAT question types. 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