Understanding Sequences: Definitions, Types, and Formulas
This guide provides an overview of sequences in mathematics, focusing on both arithmetic and geometric types. It explains key concepts such as positive integers, domain, and functions. The text details explicit and recursive definitions, highlighting how to formulate both types of sequences. For arithmetic sequences, it discusses common differences and provides formulas for both recursive and explicit representations. Geometric sequences are similarly defined, with an emphasis on common ratios and their formulas. Examples are provided to clarify these concepts.
Understanding Sequences: Definitions, Types, and Formulas
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Presentation Transcript
Sequences 2, 4, 6, 8, 10, ….
Vocabulary • A Sequence is a function whose domain is the set of positive integers. {1, 1, 2, 3, 5, 8, …} • A Function is a relation where for each input there is exactly one output. • Domain is the set of all inputs in a function. • Positive integers : {1, 2, 3, 4, 5, …} • An explicit definition of a sequence tells us what the nth term of a sequence will be, given n. • i.e: an = 2n+1 • A recursive definition of a sequence tells us the next term of a sequence given the term before it. You always have to indicate the first term of the sequence. • i.e.: a1 = 3 an = an-1 +2
Arithmetic Sequences • When the difference between successive terms of a sequence is always the same number, the sequence is called arithmetic. • The recursive formula for an arithmetic sequence is: • a1 = a, an= an-1 + d, where d is the common difference, and a is the first term in the sequence. • The explicit formula for an arithmetic sequence is: • an = d(n-1)+a1 , where d is the common difference and a1 is the first term in the sequence.
Example • Determine if the following sequence is arithmetic, if it is find the common difference and write a recursive formula and explicit formula for the sequence. • {2, 4, 6, …} • {3, 6, 9, …}
Geometric Sequences • When the ratio (divide) of successive terms of a sequence is always the same nonzero number, the sequence is called geometric. • The recursive formula for a geometric sequence is: • a1 = a, an= ran-1, where r is the common ratio • The explicit formula for a geometric sequence is: • an = a1rn-1, where r is the common ratio and a1 is the first term in the sequence.
