Download 10-1 Sequences, Series, and Sigma Notation * A sequence is an

PreCalculus
10-1 Sequences, Series, and Sigma Notation
Name:___________________
* A sequence is an ordered list of numbers. Each number in the sequence is known as a term.
You can define a sequence as a function whose domain is a set of consecutive integers. If the
domain is not specified, it starts with 1. The values of the range are called the terms of the
sequence.
Domain: 1 2 3 4 … n
Range: a1 a2 a3 a4 … an
its position of each term
term values of the sequence
* A finite sequence has a limited number of terms.
* An infinite sequence continues without stopping (…).
* A sequence can be specified by an equation, or rule.
Ex. 1: Write the first four terms of (a) an  2n(1) n and (b) f (n)  n3  10 .
* If the terms of the sequence have a unique, recognizable pattern, you may be able to write a
rule for the nth term an as a function of n (=explicit formula).
* Sequence can also be defined recursively. Recursively defined sequences give one or more of
the first few terms and then define the terms that follow using those previous terms. In this case,
the formula defining the nth term of the sequence is called a recursive formula.
*The Fibonacci sequence describes many patterns found in nature. This sequence is often
defined recursively (see p.591 Example 3).
Ex. 2: Describe the pattern, and find the next four terms of the sequence.
(a) 2, 5, 10, 17, …
(b) 32, 16, 8, 4, …
Ex. 3: Find the fourth term of the recursively defined sequence a1  3, an  3an1  n  2, where n  2.
PreCalculus Ch 10 Notes_Page 1
You Try: Find the sixth term of each sequence.
(a) a1  3, an  (2)an1 , where n  2
(b) a1  8, an  2an1  n, where n  2
*Recall the end behavior of the graph of functions. If a sequence has a limit such that the terms
approach a unique number, then it is convergent. If not, the sequence is divergent.
*Ex) convergent: an  0 as n increases.
divergent: an   as n increases.
* To graph or make a table of a sequence, let the horizontal axis represent the position numbers
(n is input, x) and the vertical axis represent the terms ( an is output, y).
Ex. 4: Determine whether each sequence is convergent or divergent. Then justify your answer using
a table, graph or rule.
 1 
(a) an  4000  
 4 
n
(b) an 
(2)n
2
(c) an  2n  5
(d) a1  36, an 
1
an 1 , where n  2
2
* Series and Summation Notation: When the terms of a sequence are added together, the
resulting expression is a series. Series can be finite or infinite. The sum of the first n terms of
a series is called the n-th partial sum and is denotes S n .
6
Ex. 5: Find fifth partial sum.
You Try: Find S 4 of an  n .
2
1
(a) an  n 2  3 and
(b) an  3( ) n
10
PreCalculus Ch 10 Notes_Page 2
You can use summation notation to write a series (= Sigma Notation)
4
(Ex.) 2  4  6  8   2i read as “the sum of 2i for values of i from 1 to 4.”
i 1
The index for summation is i . The lower bound of summation is 1 and the upper bound is 4.
Index for Summation can be any letter (doesn’t have to be i) and it can start with any number.
Ex. 6: Write the series using summation notation.
(a) 5 10 15 
100
(b)
1 4 9 16
   
2 5 10 17
Ex. 7: Find the sum. State if each series is finite or infinite.
4
5
3i  3
(a)  (n 2  n)
(b) 
i
n 1
i2
5
You Try: Find the sum of the series
 (3k
2

(c)
1
 3(10
k 1
You Try: Find the sum of the series:
5
 8i
i 1
PreCalculus Ch 10 Notes_Page 3
k
7
(b)
k 3
)
 7) . State the upper and lower bound of the summation.
k 3
(a)
k
2
 1
34
(c)
2
i1
Pre-Calculus
10-2 Arithmetic Sequences and Series
Name:__________________
*Arithmetic Sequence is a sequence in which the difference between successive terms is
constant. This constant difference is called the common difference, d.
* d  a2  a1 or an  an1 : (any term  previous term)
Ex. 1: Determine the common difference and find the next four terms of the arithmetic sequence.
(a) 7, 13, 19, 25, …
(b) -129, -98, -67, …
*Rule of an Arithmetic Sequence: The nth term of an arithmetic sequence with first term a1
and common difference d is given by: an  a1   n 1 d
*When you write a rule, you need to keep an and n.
Just find a1 and d values, then plug them in the formula.
Ex) The nth term of an arithmetic sequence with first term 2 and common difference 3 is given
by: an  2   n 1 3 or an  1  3n
Ex) The 16th term of 2, 5, 8, … is an  2  16 1 3 or 47.
Ex. 2: Find both an explicit formula and a recursive formula for the nth term of the arithmetic
sequence 14, 3, -8, …
You Try: Write a rule for the nth term of the sequence. Then find a20.
(a) -7, -10, -13, -16, …
(b) 11, 4, -3, -10, …
Ex. 3: One term of an arithmetic sequence is a27 = 263. The common difference is d = 11.
Find the first term and write a rule for the nth term.
PreCalculus Ch 10 Notes_Page 4
You Try: Find d of the arithmetic sequence for which a1  75 and a38  56.5
*If two nonconsecutive terms of an arithmetic sequence are known, the terms between them can be
calculated. These terms are called arithmetic means.
Ex. 4: Write an arithmetic sequence that has five arithmetic means between 6.2 and -1.6.
You Try: Write an arithmetic sequence that has six arithmetic means between 12.4 and -24.7.
*Second differences and quadratic functions: Subtracting a term of the sequence from the following
term gives the first differences of the sequence. The second differences are found by subtracting the
resulting first differences. When the second differences are constant, the sequence can be modeled by a
quadratic function ( an  an 2  bn  c ). When this is the case, the first three terms of the sequence can be
used to create a system of three equations to find the values of a, b, and c. Then use rref or inverse matrix
to solve the system.
Ex. 5: Find a quadratic model for the sequence -14, -8, 0, 10, 22, 36, …
You Try: Find a quadratic model for the sequence 0, 5, 12, 21, 32, 45, …
PreCalculus Ch 10 Notes_Page 5
*An Arithmetic Series is the expression formed by adding the terms of an arithmetic sequence.
The sum of the first n terms of an arithmetic series is denoted by Sn.
*The Sum of a Arithmetic Series: The sum of the first n terms of an arithmetic series is:
n
(a1  an )
2
n
or Sn  (2a1  (n  1)d )
2
Sn 
 n : # of terms

a : first term value 
when  1

a n : last term value 
Ex. 6: Find the indicated sum of each arithmetic series.
28
(a)
  2  4i 
i 1
(b) 3  2  7  12    157
32
(c)
 i  6
i 10
(d) the 17th partial sum of the arithmetic series 53  31  9 
Ex. 7: An artist receives royalties for all CDs of his music that are sold. For the first 10,000 CDs sold, he
receives $10,000. He receives $12,500 for the next 10,000 CDs, $15,000 for the next 10,000 CDs, and so
on. How much will he receive if 500,000 of his CDs are sold?
PreCalculus Ch 10 Notes_Page 6
Pre-Calculus
10.3 Geometric Sequences and Series
Name:__________________
A sequence in which the ratio between successive terms is a constant is called a geometric
sequence. The constant is referred to as the common ratio and denoted r. You can find the
common ratio by dividing any term in the sequence by the term before it. To find the next term in
the sequence, multiply a term by the common ratio.
Ex. 1: Determine the common ratio, and find the next three terms of each geometric sequence.
a)  6, 9,  13.5, …
2) (243n  729), (  81n+243), (27n  81),…
Geometric sequences can be written using an explicit formula (can tell you any given term
number), or a recursive formula (requires the knowledge of a prior term).
An explicit formula can be used to find any term in a sequence.
Explicit formula : an  a1r n 1
Recursive formula : an  ran1
Ex. 2: Write an explicit and recursive formula for finding the nth term in the sequence.
a.  1, 2,  4…
b. 2, 25, 312.5…
Ex. 3: Find the given term in the geometric sequence.
a. 11th term:  122, 115.9,  110.105,…
b. a12 if a3 = 32 and r =  4
Ex. 4: One term of a geometric sequence is a5 = 54. The common ratio is r = 3.
Find an explicit formula for the nth term.
PreCalculus Ch 10 Notes_Page 7
Ex. 5: Two terms of a geometric sequence are a3 = 45 and a5 = 405. Find an explicit formula for the nth
term.
You Try: Write a rule for the nth term of the geometric sequence. Then find a8.
a) a6  96 , r = 2
b) a 3  24, a 6  3
Ex. 6: A couple purchased a home for $225,000. After each year, the value of the home appreciates 3%.
a. Write an explicit formula for finding the value of the home after n years.
b. What is the value of the home after ten years?
Similar to arithmetic sequences, if two nonconsecutive terms of a geometric sequence are known, the
terms between them can be calculated. There are called geometric means.
Ex. 7: Write a sequence using geometric means.
a) 3 geometric means between 264 and 1.03125
PreCalculus Ch 10 Notes_Page 8
b) 2 geometric means between -4 and 13.5
A Geometric Series is the sum of the terms in a geometric sequence.
Sn 
a1 (1  r n )
1 r
or
Sn 
a1  an r
1 r
Ex. 8: Find the sum of the following geometric series.
a. the sum of the first eleven terms
4,  6, 9,…
b. the sum of the first n terms
a1 =  4, an =  65,536, and r = 2
Ex. 9: Find the sum using Sigma notation.
8
a.
 2(2)
n 1
n 3
11
b.
1
120( )

2
n4
n 1
Ex. 10: In 1999 about 586 thousand used cars were sold in Maryland. From 1999 through 2004, the
number of used cars sold increased by 3.8% per year. (Let n = 1 represent 1999)
(a) Write a rule for the number of used cars sold an (in thousands) in terms of the year.
(b) What was the total number of used cars sold in Maryland from 1999 to 2004?
PreCalculus Ch 10 Notes_Page 9
You Try: A company plans to increase production of a new item by 10% each year over the next 12
years. The company will produce 70,000 units next year (year of 1).
(a) Write a rule giving the number of units produced by the company in year n.
(b) Find the numbers of units produced in the year of 4th and 12th.
(c) Find the total number of units produced over the next 12 years.
*Sum of Infinite Series: If an infinite geometric series has a common ratio (r),
such that -1< r < 1, then the geometric series has a finite sum given by:
S 
a1
when r  1 .
1 r
* If |r| > 1, the series has no sum.
Ex. 11: Find the sum of the infinite geometric series, if possible.

1
a.  3  
i 1  4 
i 1
1 1 1 1
b.     
2 4 8 16
c. 1 
3 9 27
 

2 4 8

d.
 3  0.65
n 1
n2
You Try: Find the sum of the geometric series, if it exists.

 1
a.    
2
n 1 
n 1

3
b.   
i 0  4 
PreCalculus Ch 10 Notes_Page 10
i
3 3 3
c. 3    
4 16 64
d. .33 + .66 + 1.32 + …