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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

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sequence
Sequence:
  • A list of ordered numbers separated by commas.
  • Each number in the list is called a term.
  • For Example:

Sequence 1Sequence 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)

slide3
Sequence 1Sequence 2

2,4,6,8,10 2,4,6,8,10,…

A sequence can be finite or infinite.

The sequence has a last term or final term.

(such as seq. 1)

The sequence continues without stopping.

(such as seq. 2)

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

4,3,2,1,0,-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

2,4,8,16,32,64

Examples:
examples write a rule for the nth term
The seq. can be written as:

Or, an=2/(5n)

The seq. can be written as:

2(1)+1, 2(2)+1, 2(3)+1, 2(4)+1,…

Or, an=2n+1

Examples: Write a rule for the nth term.
example write a rule for the nth term
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
Graphing a Sequence
  • Think of a sequence as ordered pairs for graphing. (n , an)
  • 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.

Term #

Actual term

series
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 SeriesInfinite Series

2+4+6+8+10 2+4+6+8+10+…

summation notation
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 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”

i goes from 1

to 5.

summation notation for an infinite series
Summation Notation for an Infinite Series
  • Summation notation for the infinite series:

2+4+6+8+10+… would be written as:

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.

Notice the series can be written as:

Examples: Write each series in summation notation.
example find the sum of the series
Example: Find the sum of the series.
  • 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

example find the sum
Example: Find the sum.
  • Use the 3rd shortcut!
arithmetic sequence
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 (commondifference is 6)

Example: Decide whether each sequence is arithmetic.
example write a rule for the nth term of the sequence 32 47 62 77 then find a 12
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

slide20
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 a n 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 a 5 10 and a 30 110 write a rule for the nth term
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

arithmetic series
Arithmetic Series
  • The sum of the terms in an arithmetic sequence
  • The formula to find the sum of a finite arithmetic series is:

Last Term

1st Term

# of terms

example consider the arithmetic series 20 18 16 14
Find the sum of the 1st 25 terms.

First find the rule for the nth term.

an=22-2n

So, a25 = -28 (last term)

Find n such that Sn=-760

Example: Consider the arithmetic series 20+18+16+14+… .
slide26
-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 sequence
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

Example: Decide whether each sequence is geometric.
rule for a geometric sequence
Rule for a Geometric Sequence

an=a1rn-1

  • Example: Write a rule for the nth term of the sequence 5, 2, 0.8, 0.32,… . Then find a8.
  • First, find r.
  • r= 2/5 = .4
  • an=5(.4)n-1

a8=5(.4)8-1

a8=5(.4)7

a8=5(.0016384)

a8=.008192

slide31
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)

1/9=a1

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: 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.
example two terms of a geometric sequence are a 2 4 and a 6 1024 write a rule for the nth term
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/r=a1

-1024=(-4/r)r5

-1024=-4r4

256=r4

4=r & -4=r

If r=4, then a1=-1.

an=(-1)(4)n-1

If r=-4, then a1=1.

an=(1)(-4)n-1

an=(-4)n-1

Both Work!

formula for the sum of a finite geometric series
Formula for the Sum of a Finite Geometric Series

n = # of terms

a1 = 1st term

r = common ratio

p 675

p.675

Infinite Geometric Series

example an infinite geom series has a 1 4 a sum of 10 what is the common ratio
Example: An infinite geom. Series has a1=4 & a sum of 10. What is the common ratio?

10(1-r)=4

1-r = 2/5

-r = -3/5

example write 0 181818 as a fraction
Example: Write 0.181818… as a fraction.

0.181818…=18(.01)+18(.01)2+18(.01)3+…

Now use the rule for the sum!