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Unit 2: Sequences, Series, and Finicial Applications
Terms
sequence: is an ordered set of numbres.
Arithmetic Sequences: is a sequence where the
difference between each term is constant, and the constant is known as
the common difference.
Geometric Sequences: is a sequence in which the
ratio between each term is constant, and the constant is known as the
common ratio.
Note: Not all sequences are arithmetic and geometric!
finite series: finite series have a finite number of terms. - eg. \(`1 + 2 + 3 + \cdots + 10`\).
infinite series: infinite series have infinite number of terms. - eg. \(`1 + 2 + 3 + \cdots`\)
Terms in a sequence are numbered with subscripts: \(~t_1, t_2, t_3, \cdots t_n`\) where \(`t_n`\)is the general or \(`n^{th}`\) term.
Recursion Formula
A sequence is defined recursively if you have to calculate a term in a sequence from previous terms. The recursion formula consist of 2 parts.
- Base term(s)
- A formula to calculate each successive term.
eg. \(`t_1 = 1, t_n = t_{n-1} + 1 \text{ for } n \ge 1`\)