![]() ![]() ![]() 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. It uses the Greek symbol for the letter sigma.Īn infinite sequence should not be confused with an infinite series, which involves adding the numbers instead of listing them.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. Another form of notation that is used with sequences is called summation or sigma notation. A geometric infinite sequence starting with 2 with a common ratio of x2 would look like this: The progression of a geometric infinite sequence is marked by the “common ratio.” For example, a common ratio may indicate that each consecutive number is multiplied by 2. The interval between the terms is called the “common difference.” For instance, an arithmetic infinite sequence starting with 2 with a common difference of 2 would look like this: An arithmetic infinite sequence is a progression of numbers where the difference between each consecutive term is constant. Two types of infinite sequence deserve attention here. Using such terminology expresses a notation for infinity – even if humans do not have a full understanding. For instance, an infinite sequence of numbers may be represented this way: To try to understand something about the elusive concept of infinity, mathematicians use various forms of language and symbolism. In 1948, the computer scientist Alan Turing wrote about a machine with “an unlimited memory capacity obtained in the form of an infinite tape marked out into squares….” Despite the endless nature of the theoretical machine, it would be operated by a finite table of instructions. Humans have been trying to get a grasp on infinity since ancient times. ![]()
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