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Sequences (functions over the naturals) We denote the natural numbers: . A sequence is a function from . It is like an indexed variable except that it does not necessarily terminate. We will denote a sequence a by the openended ordered collection of numbers: . The ellipsis at the end means that these numbers go on and on and do not end. An indexed variable may be thought of as a terminated (or a finite) sequence: . Notation: if f is a function from {0, 1, 2, . . .} to S we usually denote f(i) by ai and we write {a0, a1, a2, a3, . . . } = {ai}i≥0 Example: Using zero-origin indexing, if f(i) = 1/(i + 1). then the sequence f = {1, 1/'2,1/3,1/4, . . . } = {a0, a1, a2, a3, . . } Definition A sequence is a function from the Since we can list the integers 1,2,3,... we likewise list a sequence f(1), f(2), f(3), f(4), … We shall denote a sequence by an italic capital letter, the set of real values that function takes by the same non-italic capital letter , and the elements of that set with the corresponding lower case letter, and subscripts. For example sequence S takes values in the set S with elements . S is a set of reals, S is a function from the integers to the reals, two different concepts. While we are being rigorous we must be careful not to confuse the two, but in general usage the concepts are interchangable. We can also denote sequences by their function. For example if we say the function S is 3k, then the sequence consists of 3,6,9,…. A sequence is an ordered set of mathematical objects. Sequences of object are most commonly denoted using braces. For example, the symbol denotes the infinite sequence of even numbers Explicit Formula of a Sequence: A formula that allows direct computation of any term for a sequence a1, a2, a3, . . . , an, . . . . Recursive Formula: For a sequence a1, a2, a3, . . . , an, . . . a recursive formula is a formula that requires the computation of all previous terms in order to find the value of an . Note: Recursion is an example of an iterative procedure. Sequences Example: the sequence 1, 4, 9, 16,... is produced by the formula for the nth term Example: the sequence 5, 7, 12, 19, 31,... is defined by the recurrence relation below Progressions Example: the arithmetic progression with first term 6 and difference 4 (6, 10, 14, 18, 22,...) is defined by an 6 4 n 1 Example: the geometric progression with first term 3 and common ratio 2 (3, 6, 12, 24, 48,... ) is defined by an 3 2 n1