Transcript 11.2 Arithmetic Sequences & Series

An Introduction to
Sequences & Series
p. 651
Sequence:
• A list of ordered numbers separated by
commas.
• Each number in the list is called a term.
• For Example:
Sequence 1
Sequence 2
2,4,6,8,10
2,4,6,8,10,…
Term 1, 2, 3, 4, 5
Term 1, 2, 3, 4, 5
Domain – relative position of each term (1,2,3,4,5)
Usually begins with position 1 unless otherwise
stated.
Range – the actual terms of the sequence
(2,4,6,8,10)
Sequence 1
2,4,6,8,10
Sequence 2
2,4,6,8,10,…
A sequence can be finite or infinite.
The sequence has a
last term or final
term.
The sequence continues
without stopping.
(such as seq. 2)
(such as seq. 1)
Both sequences have a general rule: an = 2n where n is
the term # and an is the nth term.
The general rule can also be written in function notation:
f(n) = 2n
Examples:
• Write the first 6
terms of an=5-n.
• a1=5-1=4
• a2=5-2=3
• a3=5-3=2
• a4=5-4=1
• a5=5-5=0
• a6=5-6=-1
• Write the first 6
terms of an=2n.
• a1=21=2
• a2=22=4
• a3=23=8
• a4=24=16
• a5=25=32
• a6=26=64
• 4,3,2,1,0,-1
• 2,4,8,16,32,64
Examples: Write a rule for the nth term.
2 2 2
2
a.
, ,
,
,...
5 25 125 625
The seq. can be
written as:
2 2 2 2
, 2 , 3 , 4 ,...
1
5 5 5 5
Or,
an=2/(5n)
b. 3,5,7,9,...
• The seq. can be
written as:
2(1)+1, 2(2)+1, 2(3)+1,
2(4)+1,…
Or, an=2n+1
Example: write a rule for the nth term.
• 2,6,12,20,…
• Can be written as:
1(2), 2(3), 3(4), 4(5),…
Or,
an=n(n+1)
Graphing a Sequence
• Think of a sequence as ordered pairs for
graphing. (n , an)
Term #
Actual term
• For example: 3,6,9,12,15
would be the ordered pairs (1,3), (2,6),
(3,9), (4,12), (5,15) graphed like points in a
scatter plot
* Sometimes it helps to find the rule first
when you are not given every term in a
finite sequence.
Series
• The sum of the terms in a sequence.
• Can be finite or infinite
• For Example:
Finite Seq.
Infinite Seq.
2,4,6,8,10
2,4,6,8,10,…
Finite Series
2+4+6+8+10
Infinite Series
2+4+6+8+10+…
Summation Notation
• Also called sigma notation
(sigma is a Greek letter Σ meaning “sum”)
The series 2+4+6+8+10 can be written as:
i goes from 1
5
 2i
to 5.
1
i is called the index of summation
(it’s just like the n used earlier).
Sometimes you will see an n or k here instead of i.
The notation is read:
“the sum from i=1 to 5 of 2i”
Summation Notation for an
Infinite Series
• Summation notation for the infinite series:
2+4+6+8+10+… would be written as:

 2i
1
Because the series is infinite, you must use i
from 1 to infinity (∞) instead of stopping at
the 5th term like before.
Examples: Write each series in
summation notation.
a. 4+8+12+…+100
• Notice the series can
be written as:
4(1)+4(2)+4(3)+…+4(25)
Or 4(i) where i goes
from 1 to 25.
25
 4i
1
1 2 3 4
b.     ...
2 3 4 5
• Notice the series
can be written as:
1
2
3
4



 ...
11 2 1 3 1 4 1
i
Or,
where i goes from 1 to .
i 1

i
1 i  1
Example: Find the sum of the
10
series.
2
k
1
5
• k goes from 5 to 10.
• (52+1)+(62+1)+(72+1)+(82+1)+(92+1)+(102+1)
= 26+37+50+65+82+101
= 361
Special Formulas (shortcuts!)
n
1  n
i 1
n(n  1)
i

2
i 1
n
n(n  1)(2n  1)
i 

6
i 1
n
2
Example: Find the sum.
10
i
2
i 1
• Use the 3rd shortcut!
n( n  1)( 2n  1)
6
10 *11 * 21

6
10(10  1)( 2 *10  1)

6
2310

 385
6
Arithmetic Sequences
& Series
Arithmetic Sequence:
• The difference between consecutive
terms is constant (or the same).
• The constant difference is also known
as the common difference (d).
(It’s also that number that you are adding
everytime!)
Example: Decide whether each
sequence is arithmetic.
•
•
•
•
•
•
•
-10,-6,-2,0,2,6,10,…
-6--10=4
-2--6=4
0--2=2
2-0=2
6-2=4
10-6=4
Not arithmetic (because
the differences are
not the same)
•
•
•
•
•
5,11,17,23,29,…
11-5=6
17-11=6
23-17=6
29-23=6
• Arithmetic (common
difference is 6)
Rule for an Arithmetic Sequence
an=a1+(n-1)d
Example: Write a rule for the nth
term of the sequence 32,47,62,77,… .
Then, find a12.
• The is a common difference where d=15,
therefore the sequence is arithmetic.
• Use an=a1+(n-1)d
an=32+(n-1)(15)
an=32+15n-15
an=17+15n
a12=17+15(12)=197
Example: One term of an arithmetic sequence
is a8=50. The common difference is 0.25.
Write a rule for the nth term.
• Use an=a1+(n-1)d to find the 1st term!
a8=a1+(8-1)(.25)
50=a1+(7)(.25)
50=a1+1.75
48.25=a1
* Now, use an=a1+(n-1)d to find the rule.
an=48.25+(n-1)(.25)
an=48.25+.25n-.25
an=48+.25n
Now graph an=48+.25n.
• Just like yesterday, remember to graph the
ordered pairs of the form (n,an)
• So, graph the points (1,48.25), (2,48.5),
(3,48.75), (4,49), etc.
Example: Two terms of an arithmetic sequence are
a5=10 and a30=110. Write a rule for the nth term.
• Begin by writing 2 equations; one for each term
given.
a5=a1+(5-1)d OR 10=a1+4d
And
a30=a1+(30-1)d OR 110=a1+29d
• Now use the 2 equations to solve for a1 & d.
10=a1+4d
110=a1+29d (subtract the equations to cancel a1)
-100= -25d
So, d=4 and a1=-6 (now find the rule)
an=a1+(n-1)d
an=-6+(n-1)(4) OR an=-10+4n
Example (part 2): using the rule an=-10+4n,
write the value of n for which an=-2.
-2=-10+4n
8=4n
2=n
Arithmetic Series
• The sum of the
terms in an
arithmetic sequence
1st Term
• The formula to find
the sum of a finite
arithmetic series is:
Last
Term
 a1  an 
S n  n

 2 
# of terms
Example: Consider the arithmetic
series 20+18+16+14+… .
• Find the sum of the 1st • Find n such that Sn=-760
25 terms.
 a1  an 
S n  n

• First find the rule for
2 

the nth term.
• an=22-2n
 20  (22  2n) 
 760  n

2
• So, a25 = -28 (last term)


 a1  an 
S n  n

 2 
 20  28 
S 25  25
 S25  25(4)  100
2


 20  (22  2n) 
 760  n

2


-1520=n(20+22-2n)
-1520=-2n2+42n
2n2-42n-1520=0
n2-21n-760=0
(n-40)(n+19)=0
n=40 or n=-19
Always choose the positive solution!
Geometric Sequences &
Series
Geometric Sequence
• The ratio of a term to it’s previous term
is constant.
• This means you multiply by the same
number to get each term.
• This number that you multiply by is
called the common ratio (r).
Example: Decide whether each
sequence is geometric.
• 4,-8,16,-32,…
• -8/4=-2
• 16/-8=-2
• -32/16=-2
• Geometric (common
ratio is -2)
• 3,9,-27,-81,243,…
• 9/3=3
• -27/9=-3
• -81/-27=3
• 243/-81=-3
• Not geometric
Rule for a Geometric Sequence
n-1
an=a1r
Example: Write a rule for the nth term of the
sequence 5, 2, 0.8, 0.32,… . Then find a8.
•First, find r.
a8=5(.4)8-1
•r= 2/5 = .4
a8=5(.4)7
•an=5(.4)n-1
a8=5(.0016384)
a8=.008192
Example: One term of a geometric sequence
is a4=3. The common ratio is r=3. Write a rule
for the nth term. Then graph the sequence.
• If a4=3, then when
n=4, an=3.
• Use an=a1rn-1
3=a1(3)4-1
3=a1(3)3
3=a1(27)
1/ =a
9
1
• an=a1rn-1
an=(1/9)(3)n-1
• To graph, graph the
points of the form
(n,an).
• Such as, (1,1/9),
(2,1/3), (3,1), (4,3),…
Example: Two terms of a geometric sequence are
a2=-4 and a6=-1024. Write a rule for the nth term.
• Write 2 equations, one for each given term.
a2=a1r2-1 OR -4=a1r
a6=a1r6-1 OR -1024=a1r5
• Use these 2 equations & substitution to solve for a1
& r.
-4/ =a
If r=4, then a1=-1.
r
1
-1024=(-4/r)r5 an=(-1)(4)n-1
If r=-4, then a1=1.
4
-1024=-4r
n-1
a
=(1)(-4)
4
n
256=r
Both
n-1
a
=(-4)
4=r & -4=r
n
Work!
Formula for the Sum of a Finite
Geometric Series
1 r
S n  a1 
1

r

n
n = # of terms
a1 = 1st term
r = common ratio



Example: Consider the geometric
series 4+2+1+½+… .
• Find the sum of the
first 10 terms.
• Find n such that Sn=31/4.
n
1 r n 


S n  a1 
 1   1 
 1 r 
31   2 
 4
10


1
1
4
 1    
1



2 
2
S10  4

1 
 1



2


1   1023

1
 

2046 1023

1024
1024
  4
  4
S10  4

1

  1   1024 128

 

2

  2 







  1 n 
1   
31   2  
 4
1 
4
 1



2


31
1
 1  
32
 2
32  2
n
n
  1 n 
1   
31   2  
 4

1
4




2


1
1
  
32
2
n
log232=n
n

31   1  
 8 1  
 2 
4


1 1
 
32  2 
n5
n
1 1n
 n
32 2
Infinite Geometric Series
p.675
The sum of an
infinite geometric
series
a1
S
, if r  1
1 r
If r  1, thereis no sum.
Example: Find the sum of the
infinite geometric series.

 2(0.1)
i 1
For this series, a1=2 & r=0.1
2
S
1  .1
2 20
 
.9 9
i 1
Example: Find the sum of the
4
4
series: 12  4    ...
So, a =12
1
and
3
r=1/3
9
12
S
2
3
12
S 
1
1
3
36
S
2
S=18
Example: An infinite geom. Series has a1=4 &
a sum of 10. What is the common ratio?
4
10 
1 r
10(1-r)=4
1-r = 2/5
-r = -3/5
a1
use S 
1 r
3
r
5
Example: Write 0.181818… as a
fraction.
0.181818…=18(.01)+18(.01)2+18(.01)3+…
Now use the rule for the sum!
a1

1 r
.18

1  .01
.18

.99
2

11