2.8 KiB
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.
Series: A series is the sum of the terms of a sequence.
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. \(`t_1 = 1, t_n = t_{n-1} + 1 \text{ for } n \gt 1`\)
Aritmetic Sequences
Basically, you add the commmon difference to the current term to get the next term. As such, it follows the following pattern:
\(`a, a+d, a+2d, a+3d, a+4d, \cdots`\). Where \(`a`\) is the first term and \(`d`\) is the common difference.
As such, the general term of the aritmetic sequence is:
\(`\large t_n = a + (n - 1)d`\)
Geoemetric Sequences
Basically, you multiply by the common ratio to the current term toget the next term. As such, it follows the following pattern:
\(`a, ar, ar^2, ar^3, ar^4, c\dots`\). Where \(`a`\) is the first term and \(`r`\) is the common ratio.
As such, the general term of the geometric sequence is:
\(`\large t_n = a(r)^{n-1}`\)
Aritmetic Series
An arithmetic series is the sum of the aritmetic sequence’s terms.
The formula to calculate is:
\(`\large S_n = \dfrac{n(a_1 + a_n)}{2}`\) Or \(`\large S_n = \dfrac{n(2a_1 + (n-1)d)}{2}`\)
Geometric Series
- A geoemtric series is created by adding the terms of the geometric sequence.
The formula to calulate the series is:
\(`\large S_n= \dfrac{a(r^n- 1)}{r-1}`\) or \(`\large S_n = \dfrac{a(1 - r^n)}{1 - r}`\)
Series and Sigma Notation
Its often convient to write summation of sequences using sigma notation. In greek, sigma means to sum.
eg. \(`S_ = u_1 + u_2 + u_3 + u_4 + \cdots + u_n = \sum_{i=1}^{n}u_i`\)
\(`\sum_{i=1}^{n}u_i`\) means to add all the terms of \(`u_i`\) from \(`i=1`\) to \(`i=n`\).
Programmers might refer to this as the for loop.
int sum=0;
for(int i=1; i<=N; i++) {
sum += u[i];
}Infinite Geometric Series
Either the series converges and diverges. There is only a finite sum when the series converges.
Recall the