The Fibonacci sequence is named after Leonardo Fibonacci, a famous Italian mathematician. Such a type of sequence is called the Fibonacci sequence. Using the above formula, we can determine any number of any given geometric sequence.īy adding the value of the two terms before the required term, we will get the next term. Here, we are multiplying it by 4 every time to get the next term. Using the above formula, we can successfully determine any number of any given arithmetic sequence.Ī sequence in which every successive term has a constant ratio between them is called Geometric Sequence.Ĭonstant ratio means that between every two numbers of the geometric sequence, there is an arbitrary constant, which is multiplied by the last number of the sequence to obtain the next number. In an arithmetic sequence, if the first term is a 1 and the common difference is d, then the nth term of the sequence is given by: The difference between the two successive terms is In the above example, we can see that a 1 =3 and a 2 = 6. When we have to get the next number of this sequence, we simply add 3 to the last number of this sequence. Here, the difference between the two successive terms is 3. And if we need to generate the next number, we simply add this arbitrary constant value again to the last number of the sequence and get a new number to extend the sequence. This means that as we go further up in the sequence, the numbers keep increasing by an arbitrary constant value. There are mainly three types of sequences:Īny sequence in which the difference between every successive term is constant is called Arithmetic Sequences. Let us study the sequence and series formula. All of these calculations are similar to studying numeric patterns and extending them or summing them up to visualise a future score, which is some steps further in the extension of the sequence that was observed from past scores. One simple example is score prediction, required run rate, projected score, etc. Sequences and series are immensely useful when trying to do predictive or projective calculations. We have to just put the values in the formula for the series. Suppose we have to find the sum of the arithmetic series 1,2,3,4. Series and sequence are the concepts that are often confused. Whereas, series is defined as the sum of sequences, which means that if we add up the numbers of the sequence, then we get a series.Įxample: 1+2+3+4+.+n, where n is the nth term. x n, where 1,2,3 are the positions of the numbers and n is the nth term. ![]() We can commonly represent sequences as x 1, x 2 ,x 3. This mathematical representation of such patterns is studied under sequence and series.Īny pattern when laid out in numbers and separated by commas is known as a sequence. All of these can be determined and represented mathematically. Graphs, geometry, mandalas, snail shells, flower petals, and so on. Think of patterns that you see around you in daily life. ![]() We will familiarize you with these by giving you five mini-projects and some related problems associated with the concepts afterwards.We can define a sequence as an arrangement of numbers in some definite order according to some rule. ![]() 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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