Download 12-1 Define and Use Sequences and Series

12-1 Define and Use Sequences and Series
Name:__________________
Objective: To recognize and write rules for number patterns.
Algebra 2 Standards: 22.0 and 23.0
Alg 2
* A sequence is 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
relative 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 five terms of (a) an = 5n – 1 and
(b) f(n) = 2n – 3.
* If the terms of the sequence have a recognizable pattern, you may be able to write a rule for the
nth term of the sequence.
Ex. 2: Describe the pattern, write the next term, and write a rule for the nth term of the sequence.
(a) 2, 5, 10, 17, …
(b) -4, -8, -12, -16, -20, …
* To graph a sequence, let the horizontal axis represent the position numbers (domain) and the
vertical axis represent the terms (range).
Ex. 3: For the sequence 1, 4, 9, 16, …, describe the pattern, write the next term, graph the first
five terms, and write the rule for the nth term.
You Try: Write the first 6 terms of the sequences: an 
n
n 1
Algebra 2 Ch.12 Notes-page 1
* Series and Summation Notation: When the terms of a sequence are added together, the
resulting expression is a series (can be finite or infinite). 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 limit of summation is 1 and the upper limit is 4.
Ex. 4: Write the series using summation notation. State if each is finite or infinite.
1 1 1
(a) 1  8  27  64   729
(b) 1    
2 4 8
You Try: Write the series using summation notation.
(a) 5 10 15 
100
(b)
1 4 9 16
   
2 5 10 17
* Index for Summation can be any letter (doesn’t have to be i) and it can start with any number.
5
Ex. 5: Find the sum of the series
 (3k
2
 7) . State the upper and lower limits of the summation.
k 3
You Try: Find the sum of the series:
5
(a)
 8i
 k
7
(b)
i 1
k 3
2
 1
34
(c)
1
i1
* Formulas for Special Series:
1) Sum of n terms of 1
2) Sum of first n terms
n
n
n  n  1
1 n
i


2
i 1
i 1
3) Sum of first n terms
n
n  n  1 2n  1
i2 

6
i 1
Ex.6: Find the sum of the series:
35
(a)
i
i 1
18
(b)
k
k 1
343
2
(c)
1
i1
Algebra 2 Ch.12 Notes-page 2
12-2 Analyze Arithmetic Sequences and Series
Name:__________________
Objective: To study arithmetic sequences and series.
Algebra 2 Standards: 22.0 and 23.0
Alg 2
*Arithmetic Sequence: A sequence in which the difference of consecutive terms is constant.
This constant difference is called the common difference, d. (d=any term– previous term)
Ex. 1: Tell whether the sequence is arithmetic.
(a) 7, 13, 19, 25, …
(b) -8, -4, 0, 6, 12, …
You Try: State whether the sequence 17, 14, 11, 8, 5, … is arithmetic. Explain why or why not.
*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
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. 2: Write a rule for the nth term of the sequence. Then find a20.
(a) -7, -10, -13, -16, …
(b) 59, 68, 77, 86, …
Ex. 3: One term of an arithmetic sequence is a27 = 263. The common difference is d = 11.
(a) Write a rule for the nth term. (b) Graph the first six terms of the sequence. What do you notice
about the slope of the line?
Ex. 4: Two terms of an arithmetic sequence are a10 = 148 and a44 = 556. Find a rule for the nth term.
Algebra 2 Ch.12 Notes-page 3
*When you write a rule, you need to keep an and n.
Just find a1 and d values, then plug them in the formula an  a1   n 1 d .
You Try: Write a rule for the nth term of the arithmetic sequence. Then find a20.
(a) 17, 14, 11, 8, …
(b) a11 = -57, d = -7
(c) a7 = 26, a16 = 71
*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:
Sn 
n
(a1  an )
2
28
Ex. 5: What is the sum of the arithmetic series
  2  4i 
i 1
a.
b.
c.
d.
14
420
870
1568
Ex. 6: A tall brick building has a decoration as shown on the board.
(a) Write a rule for the number of bricks in the nth row if the top row is 1.
(b) What is the total number of bricks if the bottom row has 44 bricks?
12
You try: Find the sum of the arithmetic series
  2  7i 
i 1
Algebra 2 Ch.12 Notes-page 4
12.3 Analyze Geometric Sequences and Series
Name:__________________
Objective: To study geometric sequences and series
Algebra 2 Standards 22.0 and 23.0
*Geometric Sequences have a ratio of any term to the previous term that is constant.
This constant ratio is called the common ratio, r.
Ex. 1: Tell whether the sequence is geometric.
a) 2, 6, 18, 54, 162, …
2) 2, 7, 12, 17, 2, …
*Rule for Geometric Sequence: The nth term of a geometric sequence with first term a1 and
common ratio r is given by: an  a1r n 1
Ex. 2 Write a rule for the nth term of the sequence. Then find a6 .
a) 3, 12, 48, 192, …
4
b) 36,  12, 4,  ,
3
Ex. 3: One term of a geometric sequence is a5 = 54. The common ratio is r = 3.
(a) Write a rule for the nth term. (b) Graph the sequence.
Ex. 4: Two terms of a geometric sequence are a3 = 45 and a5 = 405. Find a rule for the nth term.
You Try: Write a rule for the nth term of the geometric sequence. Then find a8.
a) 3, 15, 75, 375, …
b) a6  96 , r = 2
c) a 3  24, a 6  3
Algebra 2 Ch.12 Notes-page 5
*Geometric Series: expression formed by adding the terms of a geometric sequence.
The sum of the first n terms of a geometric series with common ratio ≠ 1 is:
a (1  r n )
Sn  1
1 r
3
Ex. 5: Find the sum of the geometric series  4  
i 1  2 
6
i 1
How many terms are in the series?
Ex. 6: In 1999 about 586 thousand used cars were sold in Maryland. From 1999 through 2004, the
number of used cars sold increased by about 3.8% per year. (Let n = 1 represent 1999)
(a) Write a rule for the total 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?
8
You Try: Find the sum of the geometric series
 6  2 
i 1
i 1
You Try: A company plans to increase production of a product by 10% each year over the next 12
years. The company will produce 70,000 units next year (year 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 years 4, 8, and 12.
(c) Find the total number of units produced over the next 12 years.
Algebra 2 Ch.12 Notes-page 6
12.4 Find sums of Infinite Geometric Series
Name:__________________
Objective: To find the sums of geometric series.
Algebra 2 Standards 22.0 and 23.0
A2
*The sum Sn of the first n terms of an infinite series is called a partial sum.
2 2 2 2
 
Ex. 1: Consider the infinite geometric series 2   
3 9 27 81
1, 2, 3, 4, and 5. Then describe what happens to Sn as n increases.
Find the partial sums Sn for n =
*Sum of Infinite Series: The sum of an infinite geometric series with first term a1 and
common ratio r is given by S 
a1
when r  1 .
1 r
* If |r| > 1, the series has no sum. (= the sum does not exist)
Ex. 2: Find the sum of the infinite geometric series.

a.
1
3 

i 1  4 
i 1
1 1 1 1
b.     
2 4 8 16
3 9 27

Ex. 3: What is the sum of the geometric series 1   
2 4 8
A. -2
B. - ½
C. 2
D. Does not exist
?
Algebra 2 Ch.12 Notes-page 7
You Try: Find the sum of the geometric series, if it exists.

a.
 1

 
2
n 1 
n 1

b.
5
3 

i 1  4 
i 1
3 3 3
c. 3    
4 16 64
Ex. 4: A pendulum that is released to swing freely travels 15 inches on the first swing. On each
successive swing, the pendulum travels 70% of the distance of the previous swing. What is the total
distance the pendulum swings?
Ex. 5: Write the repeating decimal as a fraction in lowest terms using an infinite series.
a. 0.55555…
b. 0.467676767….
You Try: Write the repeating decimal as a fraction in lowest terms an infinite series.
a. 0.727272…
b. 5.146146146…
Algebra 2 Ch.12 Notes-page 8