PreCalculus B 2nd Hour Spring 11: Chapter 9: Sequences, Series, and Probability

An infinite sequence is a function whose domain is the set of positive integers. The function values

are the terms of the sequence. If the domain of the function consists of the first n positive integers only, the sequence is a finite sequence.

Some sequences are defined recursively. To define a sequence recursively, you need to be given one or more of the first few terms. The Fibonacci sequence is a good example

Ex.

, where

The first six terms of this sequence are

given

use recursive formula

use recursice formula

use recursive formula

Factorial: If n is a positive integer, n factorial is defined by

$n!=1\cdot 2\cdot 3\cdot 4....(n-1)\cdot n$

****As a special case, zero factorial is defined as $0!=1$ ****

Evaluating Factorial Expressions

Ex.

Summation Notation: The sum of the first n terms of a sequence is represented by

$\sum_{i=1}^{n} a_{i}=a_{1}+a_{2}+a_{3}....+a_{n}$

where i is called the index of summation, n is the upper limit of summation, and 1 is the lower limit of summation.

Ex. $\sum_{k=3}^{6}(1+k^{2})=(1+3^{2})+(1+4^{2})+(1+5^{2})+(1+6^{2})$

$=10+17+26+37$

$=90$

Properties of Sums

1. $\sum_{i=1}^{n}ca_{1}=c\sum_{i=1}^{n}a_{1}$ , c is any constant

2. $\sum_{i=1}^{n}(a_{i}+b_{i})=\sum_{i=1}^{n}a_{i}+\sum_{i=1}^{n}b_{i}$

3. $\sum_{i=1}^{n}(a_{i}-b_{i})=\sum_{i=1}^{n}a_{i}-\sum_{i=1}^{n}b_{i}$

Arithmetic Sequence: A sequence is arithmetic if the differences between consecutive terms are the same. So, the sequence

$a_{1},a_{2},a_{3},a_{4}....a_{n},....$

is arithmetic if there is a number d such that

$a_{2}-a_{1}=a_{3}-a_{2}=a_{4}-a_{3}=...=d.$

The number d is the common difference of the arithmetic sequence.

*The nth term of an arithmetic sequence has the form

$a_{n}=dn+c$

where d is the common difference between consecutive terms of the sequence and $c=a_{1}-d.$

$graph of an arithmetic sequence$

The Sum of a Finite Arithmetic Sequence

The sum of a finite arithmetic sequence with n terms is

$S_{n}=\frac{n}{2}(a_{1}+a_{n}).$

Finding a Partial Sum of an Arithmetic Sequence

Find the 150th partial sum of the sequence

5, 16, 27, 38, 49....

Solution

For this arithmetic sequence, you have $a_{1}=5$ and $d=16-5=11.$ So,

$c=a_{1}-d=5-11=-6$

and the nth term is

$a_{n}=11n-6$ .

Therefore, $a_{150}=11(150)-6=1,644,$ and the sum of the first 150 terms is

$S_{n}=\frac{150}{2}(5+1644)$

$S_{n}=75(1649)$

$S_{n}=123,675$

Geometric Sequence: A sequence is geometric if the ratios of consecutive terms are the same.

$\frac{a_{2}}{a_{1}}=\frac{a_{3}}{a_{2}}=\frac{a_{4}}{a_{3}}....=r,$ $r\neq 0$

The number r is the common ratio of the sequence

*The nth term of a geometric sequence has the form

$a_{n}=a_{1}r^{n-1}$

where r is the common ratio of consecutive terms of the sequence. So, every geometric sequence can be written in the following form

$a_{1},a_{1}r,a_{1}r^{2},a_{1}r^{3},a_{1}r^{4}....a_{1}r^{n-1},....$

*The sum of the geometric sequence

$a_{1},a_{1}r,a_{1}r^{2},a_{1}r^{3},a_{1}r^{4}....a_{1}r^{n-1},....$

with common ratio $r\neq 1$ is

$S_{n}=a_{1}(\frac{1-r^{n}}{1-r}).$

The Sum of an Infinite Geometric Series

If $\left | r \right |< 1,$ then the infinite geometric series

$a_{1},a_{1}r,a_{1}r^{2},a_{1}r^{3},a_{1}r^{4}....a_{1}r^{n-1},....$

has the sum

$S=\frac{a_{1}}{1-r}.$

PreCalculus B 2nd Hour Spring 11

Tuesday, June 14, 2011

Chapter 9: Sequences, Series, and Probability

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