![]() ![]() At the end of the first year you will have a total of: \ With simple interest, the key assumption is that you withdraw the interest from the bank as soon as it is paid and deposit it into a separate bank account. You are paid $15\%$ interest on your deposit at the end of each year (per annum). We refer to $£A$ as the principal balance. ![]() Simple and Compound Interest Simple Interest For example, \ so the sequence is neither arithmetic nor geometric. A series does not have to be the sum of all the terms in a sequence. The starting index is written underneath and the final index above, and the sequence to be summed is written on the right. We call the sum of the terms in a sequence a series. The Summation Operator, $\sum$, is used to denote the sum of a sequence. If the dots have nothing after them, the sequence is infinite. If the dots are followed by a final number, the sequence is finite. Note: The 'three dots' notation stands in for missing terms. is a finite sequence whose end value is $19$.Īn infinite sequence is a sequence in which the terms go on forever, for example $2, 5, 8, \dotso$. For example, $1, 3, 5, 7, 9$ is a sequence of odd numbers.Ī finite sequence is a sequence which ends. Contents Toggle Main Menu 1 Sequences 2 The Summation Operator 3 Rules of the Summation Operator 3.1 Constant Rule 3.2 Constant Multiple Rule 3.3 The Sum of Sequences Rule 3.4 Worked Examples 4 Arithmetic sequence 4.1 Worked Examples 5 Geometric Sequence 6 A Special Case of the Geometric Progression 6.1 Worked Examples 7 Arithmetic or Geometric? 7.1 Arithmetic? 7.2 Geometric? 8 Simple and Compound Interest 8.1 Simple Interest 8.2 Compound Interest 8.3 Worked Examples 9 Video Examples 10 Test Yourself 11 External Resources SequencesĪ sequence is a list of numbers which are written in a particular order. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License. Use the information below to generate a citation. Then you must include on every digital page view the following attribution: If you are redistributing all or part of this book in a digital format, Then you must include on every physical page the following attribution: If you are redistributing all or part of this book in a print format, Want to cite, share, or modify this book? This book uses the Round to the nearest thousandth when necessary. Use the right arrow key to scroll through the list of terms.įor the following exercises, use the steps above to find the indicated terms for the sequence. ![]() Press to see the list of terms for the finite sequence defined.See the instructions above for the description of each item. Enter the items in the order “Expr”, “Variable”, “start”, “end” separated by commas.(b) Is 100 a term of this sequence Why (c) Prove that the square of any term of this. Use the right arrow key to scroll through the list of terms. (a) Write the algebraic form of the arithmetic sequence 1,4,7,10. Press to see the list of terms for the finite sequence defined. You will see the sequence syntax on the screen. Press 3 times to return to the home screen.In the line headed “end:” key in the value of n n that ends the sequence.In the line headed “start:” key in the value of n n that begins the sequence.In the line headed “Variable:” type in the variable used on the previous step.In the line headed “Expr:” type in the explicit formula, using the button for n n.Scroll over to OPS and choose “seq(” from the dropdown list.For example, suppose we know the following:įind the tenth term of the sequence a 1 = 2 a 1 = 2, a n = n a n − 1 a n = n a n − 1įollow these steps to evaluate a finite sequence defined by an explicit formula. Instead, we describe the sequence using a recursive formula, a formula that defines the terms of a sequence using previous terms.Ī recursive formula always has two parts: the value of an initial term (or terms), and an equation defining a n a n in terms of preceding terms. The Fibonacci sequence cannot easily be written using an explicit formula. Other examples from the natural world that exhibit the Fibonacci sequence are the Calla Lily, which has just one petal, the Black-Eyed Susan with 13 petals, and different varieties of daisies that may have 21 or 34 petals.Įach term of the Fibonacci sequence depends on the terms that come before it. Their growth follows the Fibonacci sequence, a famous sequence in which each term can be found by adding the preceding two terms. We may see the sequence in the leaf or branch arrangement, the number of petals of a flower, or the pattern of the chambers in a nautilus shell. Sequences occur naturally in the growth patterns of nautilus shells, pinecones, tree branches, and many other natural structures. Writing the Terms of a Sequence Defined by a Recursive Formula ![]()
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