Then it says, on the nextīounce, the stretch is 60% of the original jump, and thenĮach additional bounce stretches the rope 60% of On jump one, the cord stretches 120 feet. Goes bungee jumping off of a bridge above water. The notion of a geometric sequence, and actually do a word I just want to make that clearīecause that used to confuse me a lot when I first learnedĪbout these things. It is that a series is the sum of a sequence. Plus negative 30, plus 10, plus negative 10/3, plus 10/9. A series, the most conventionalĪ geometric sequence. You might also see theĪ geometric series. Positive 10/9, right? Negative 1/3 times negativeġ0/3, negatives cancel out. Negative 30, then 10, then negative 10/3. This always used to confuse meīecause the terms are used very often in the I want to make one littleĭistinction here. So that's what people talkĪbout when they mean a geometric sequence. Going to be negative 10/3 times negative 1/3 so it's going To be 10 times negative 1/3, or negative 10/3. Which is what? That's negative 30, right? 1/3 times 90 is 30, and then you a1 is equal to 90 and yourĬommon ratio is equal to negative 1/3. You, hey, you've got a geometric sequence. So in this case, a1 is equal toĢ, and my common ratio is equal to 3. It by a common number, and that number is often called Some notation here, this is my first term. So let's say my first number isĢ and then I multiply 2 by the number 3. Where each successive number is a fixed multiple of Special progression, or a special sequence, of numbers, It's not a geometric sequence,īut it is a sequence. So for example, and this isn'tĮven a geometric series, if I just said 1, 2, 3, 4, 5. Just, what is a sequence? And a sequence is, youĬan imagine, just a progression of numbers. When someone tells you a geometric sequence. Start, just to understand what we're talking about And I have a ton of moreĪdvanced videos on the topic, but it's really a good place to Introduce you to the idea of a geometric sequence. On the other hand, if your sequence starts with a(0), you will often find your exponent needs to be n in order for your initial value to be correct for building the sequence you want. If you start at a(1), you will usually need to have your exponent as the expression n-1 to match the sequence that you are given. Whichever way you start numbering, it is always important to check that your formula for the sequence actually ends up with the sequence that you want. In the case that Sal is modeling, the first thing that happens is slightly different, so we call it "0" When we use sequences to match or model actual occurrences, it can get pretty interesting. It is usually easier for humans to keep track if the first item is called the n = 1 item. However, you can define your first term as a(0) in the same way that in a computer array, the first element is the 0th item. If you mean that the a(1) is your first term, then you cannot have a zero term. Back here where math is simpler, we do often talk about a(1) as the first item. We will familiarize you with these by giving you five mini-projects and some related problems associated with the concepts afterwards.You will find that sequences don't always start with a(1). There are many applications for sciences, business, personal finance, and even for health, but most people are unaware of these. This chapter is for those who want to see applications of arithmetic and geometric progressions to real life. Hence, these consecutive amounts of Carbon 14 are the terms of a decreasing geometric progression with common ratio of ½. Have you ever thought of how archeologists in the movies, such as Indiana Jones, can predict the age of different artifacts? Do not you know that the age of artifacts in real life can be established by the amount of the radioactive isotope of Carbon 14 in the artifact? Carbon 14 has a very long half-lifetime which means that each half-lifetime of 5730 years or so, the amount of the isotope is reduced by half. As a result, the total number of grains per 64 cells of the chessboard would be so huge that the king would have to plant it everywhere on the entire surface of the Earth including the space of the oceans, mountains, and deserts and even then would not have enough! The king was amazed by the “modest” request from the inventor who asked to give him for the first cell of the chessboard 1 grain of wheat, for the second-2 grains, for the third-4 grains, for the fourth-twice as much as in the previous cell, etc. According to the legend, an Indian king summoned the inventor and suggested that he choose the award for the creation of an interesting and wise game. One of the most famous legends about series concerns the invention of chess. Over the millenia, legends have developed around mathematical problems involving series and sequences.
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