sequences a2/trig. sequences: vocabulary sequence: an ordered list of numbers –ex. -2, -1, 0, 1,...
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Sequences
A2/trig
Sequences:Vocabulary
• Sequence: an ordered list of numbers– Ex. -2, -1, 0, 1, 2, 3
• Term: each number in a sequence– Ex. a1, a2, a3, a4, a5, a6
• Recursive Formula: finding the next term depends on knowing a term or terms before it.
• Explicit formula: defines the nth term of a sequence.
Vocabulary
• Recursive Formula: – Uses one or more previous terms to generate
the next term.
– Any term: an
– Previous term: an-1 an-1
Examples:
A) Write the first six terms of the sequence
where a1 = -2 and an = 2an-1 – 1
( Always list the terms with subscripts first:) a1, a2, a3, a4, a5, a6
B) Write the first six terms of the sequence where a1 = 4 and an = 3an-1 + 5
a1, a2, a3, a4, a5, a6
Examples:
A) Write the first six terms of the sequence
where a1 = -2 and an = 2an-1 – 1 a1, a2, a3, a4, a5, a6
-2,-5,-11,-23,-47,-95
B) Write the first six terms of the sequence where a1 = 4 and an = 3an-1 + 5
a1, a2, a3, a4, a5, a6
4, 17, 56 ,173, 524, 1577
Recursive formula depending on two previous terms:
• a1=-2, a2 = 3 ak= ak-1 + ak-2
• Find the first 6 terms.
Recursive formula depending on two previous terms:
• a1=-2, a2 = 3 ak= ak-1 + ak-2
• Find the first 6 terms.
• a1, a2, a3, a4, a5, a6
• -2, 3, 1, 4, 5, 9
Two special sequence types:
Arithmetic sequence:a sequence in which each term is found by adding
a constant, called the common difference (d), to the previous term. Geometric sequence: a sequence in
which each term is founds by multiplying a constant, (r), called a common ratio to the previous term.
Some sequences are neither of these!
Do now:
A) Find the 10th term of a1 = 7 and an = an-1 + 6
Recursive formula
Example 1:
A) Find the 10th term of a1 = 7 and an = an-1 + 6
7,13,19,25,31,37,43,49,55,61
Formula for the nth term
an = a1 + (n – 1)d
What term you are looking for
First term in the sequence
What term you are looking for
Common difference
Example:Find the 10th term of a1 = 7 and
an = an-1 + 6 Write the explicit formula
(recall a10 = 61)
an= 7+ 6(n-1)an = 7 + 6n – 6an = 6n + 1a10 = 6(10) +1 = 61
an = a1 + d(n – 1)
Vocabulary
• Arithmetic Sequence: – A sequence generated by adding “d” a constant
number to pervious term to obtain the next term.– This number is called the common difference.
• Start by asking, What is d? a2 – a1
3, 7, 11, 15, … d = 4
8, 2, -4, -10, … d = -6
Find the explicit formula for these examples:
• For Arithmetic Sequences, use the formula: – an = a1 + d(n – 1)
3, 7, 11, 15, … d = 4
8, 2, -4, -10, … d = -6
solutions:
an = a1 + d(n – 1)3, 7, 11, 15, … d = 4
an = 3 + 4(n – 1)an = 3 + 4n – 4
an = 4n – 1
8, 2, -4, -10, … d = -6
an = 8 + -6(n – 1)an = 8 + -6n + 6
an = -6n + 14
Examples when a1 is not given
A) Find the 10th term of the arithmetic sequence where a3 = -5 and a6 = 16
B) Find the 15th term of the arithmetic sequence where a5 = 7 and a10 = 22
C) Find the 12th term of the arithmetic sequence where a3 = 8 and a7 = 20
Examples when a1 is not given
A) Find the 10th term of the arithmetic sequence where a3 = -5 and a6 = 16
16- -5 =21 6-3 = 3
A)an = -19 + 7(n – 1)
B)an = -19 + 7n – 7
C) an = 7n – 26
A)A10 = 7(10)-26=44
d =213=7 a1 =−19
Examples when a1 is not given
B) Find the 15th term of the arithmetic sequence where a5 = 7 and a10 = 22
22-7 = 15 10-5 = 5
a1=-5 a2=-2 a3=1 a4=4 a5=7
B) an = -5 + 3(n – 1)
C) an = -5 + 3n – 3
D) an = 3n – 8
B)A15 = 3(15)-8=37
d =155=3
Examples when a1 is not given
C) Find the 12th term of the arithmetic sequence where a3 = 8 and a7 = 20
D = 3 a1 = 2C)an = 2 + 3(n – 1)
D)an = 2 + 3n – 3
E) an = 3n – 1
C)A12 = 3(12)-1=35
Vocabulary
• Arithmetic Means:– Terms in between 2 nonconsecutive terms– Ex. 5, 11, 17, 23, 29 11, 17, 23 are the
arithmetic means between 5 & 29
Example 3:
A) Find the 4 arithmetic means between 10 & -30
B) Find the 5 arithmetic means between 6 & 60
Example 3:
A) Find the 4 arithmetic means between 10 & -30
10, 2, -6, -14, -22 -30
d =−30−10
5=−8
Example 3:
B) Find the 5 arithmetic means between 6 & 60
6, 15, 24, 33, 42, 51, 60
60−66
=9
Geometric Sequences
multiplying
Do Now:
• Find the 5th term of a1 = 8 and an = 3an-1
• Find the 7th term of a1 = 5 and an = 2an-1
Do Now:
• Find the 5th term of a1 = 8 and an = 3an-1
• 8, 24, 72, 216, 648
• Find the 7th term of a1 = 5 and an = 2an-1
• 5, 10, 20, 40, 80, 160, 320
Vocabulary
• Geometric Sequence:– A sequence generated by multiplying a constant
ratio to the previous term to obtain the next term.– This number is called the common ratio.
• What is r?
2, 4, 8, 16, … r = 2
27, 9, 3, 1, … r = 1/3
2
1
a
ra
Explicit Formula for the nth term
an = a1rn-1
What term you are looking for
First term in the sequence
What term you are looking for
Common Ratio
Explicit geometric formulaan = a1rn-1
• Find the Explicit formula and the 5th term of a1 = 8 and an = 3an-1
• Find the Explicit formula and the 7th term of a1 = 5 and an = 2an-1
Explicit geometric formula
• Find the explicit formula and the 5th term of a1 = 8 and an = 3an-1
• an = a1rn-1
• an = 8(3)n-1 a5 = 8(3)4 = 648
• Find the Explicit formula and the 7th term of a1 = 5 and an = 2an-1
• an = a1rn-1
• an = a1rn-1 a7 = 5(2)6 = 320
Warm up1. Find the 8th term of the sequence defined by a1= –4 and an= an-1+ 2
2. Find the 12th term of the arithmetic sequence in which a4= 2 and a7= 6
3. Find the four arithmetic means between 6 and 26.
4. Find the 5th term on the sequence defined by a1= 2 andan= 2an-1.
Summation
4
1
12n
n
Series• Series: the sum of a sequence
– Sequence: 1, 2, 3, 4– Series: 1 + 2 + 3 + 4
• Summation Notation:
4
1
12n
nEnd number
Start number
Formula to use
Summation Notation - __________________ EX. (for the above series)
4
1
12n
n
= 2(1)-1 + 2(2)-1 + 2(3) -1 + 2(4) -1
= 1 + 3 +5 + 7
=16
Summation Properties
• For sequences ak and bk and positive integer n:
1 1
1) n n
k kk k
ca c a
1 1 1
2) n n n
k k k kk k k
a b a b
Summation Formulas
• For all positive integers n:
Constant Linear
Quadratic
1
n
k
c nc
1
( 1)
2
n
k
n nk
2
1
( 1)(2 1)
6
n
k
n n nk
Example 1:
A) Evaluate
B) Evaluate
6
1
2k
k
6
1
4k
k
Example 1:
A) Evaluate
B) Evaluate
2kk=1
6
∑ =2+4 +6+8+10 +12 =42
4 kk=1
6
∑ =4(1+2 + 3+ 4 +5 +6)=84
Extra Example:
• Evaluate
(2m2 +3m+2)
m=0
2
∑
Extra Example:
• Evaluate
(2m2 +3m+2)
m=0
2
∑ =(2(0)2 +3(0)+2)+ (2(12 ) +3(1)+2)+ (2(2)2 +3(2)+2)
=2 + 7 +16
= 25
Arithmetic Series
Sum of an arithmetic sequence
1, 4, 7,10,13
9,1,−7,−15
Arithmetic Sequences
ADDTo get next term
2, 4, 8,16, 32
9,−3,1,−1 / 3
Geometric Sequences
MULTIPLYTo get next term
Arithmetic Series
Sum of Terms
Geometric Series
Sum of Terms
Do Now: add the terms of the 4 series above
1, 4, 7,10,13
9,1,−7,−15
Arithmetic Sequences
ADDTo get next term
2, 4, 8,16, 32
9,−3,1,−1 / 3
Geometric Sequences
MULTIPLYTo get next term
Arithmetic Series
Sum of Terms
35
−12
Geometric Series
Sum of Terms
62
20 / 3
Do Now: add the terms of the 4 series above
Vocabulary
• An Arithmetic Series is the sum of an arithmetic sequence.
Formula for arithmetic series
Sn=
21 naa
n
Example 1:
A. Find the series 1, 3, 5, 7, 9, 11
B. Find the series 8, 13, 18, 23, 28, 33, 38
Example 1:
A. Find the series 1, 3, 5, 7, 9, 11
B. Find the series 8, 13, 18, 23, 28, 33, 38
sn =62(1+11)=36
sn =72(8 + 38)=161
Example 2:
A) Given 3 + 12 + 21 + 30 + …, find S25
B) Given 16, 12, 8, 4, …, find S11
Find the 25th and the 11th terms by finding the explicit formula first.
Example 2:
A) Given 3 + 12 + 21 + 30 + …, find S25
B) Given 16, 12, 8, 4, …, find S11
an =3+9(n−1)=9n−6a25 =9(25)−6 =219
Now apply series formula..
Example 2:
A) Given 3 + 12 + 21 + 30 + …, find S25
B) Given 16, 12, 8, 4, …, find S11
s25 25232192775
s11 =112(16 +−24)=−44
Example 3:
A) Evaluate
B) Evaluate
12
1(6 2 )
kk
21
1(5 4 )
kk
Vocabulary
• An Geometric Series is the sum of an geometric sequence.
Formula for geometric series
Sn=
r1
r1a
n
1
Example 1:
• Given the series 3 + 4.5 + 6.75 + 10.125 + …find S5
Example 1:
• Given the series 3 + 4.5 + 6.75 + 10.125 + …find S5
s5 a11−rn
1−r
⎛
⎝⎜
⎞
⎠⎟3
1−1.55
1−1.5
⎛
⎝⎜
⎞
⎠⎟
39.5625
Vocabulary of Sequences (Universal)
1a First term
na nth term
nS sum of n terms
n number of terms
r common ratio
nth term of geometric sequence→ an =a1rn−1
sumof nterms of geometric sequence→ a1(1−rn)1−r
7
1 1 1Find S of ...
2 4 8
1a First term
na nth term
nS sum of n terms
n number of terms
r common ratio
S
n=
a1(1−rn)⎡⎣
⎤⎦
1−r
7
1 1 1Find S of ...
2 4 8
1a First term
na nth term
nS sum of n terms
n number of terms
r common ratio
1/2
7
x
NA
11184r
1 1 22 4
n1
n
a r 1S
r 1
x =
121− 1
2
⎛
⎝⎜
⎞
⎠⎟
7⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
1−12
=
121− 1
2
⎛
⎝⎜
⎞
⎠⎟
7⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
−12
127/128
Do Now:
• Evaluate
• Evaluate
4 (−5)k−1
k=1
7
∑
2+ 3k−1( )
k=1
6
∑
Do Now:
• Evaluate
• Evaluate
4 (−5)k−1
k=1
7
∑ =52,084
2+ 3k−1( )
k=1
6
∑ =2+31−1 +32−1 +33−1 +34−1 +35−1 +36−1
=366
Do Now:
• Evaluate
• Evaluate
17
14( 5)k
k
16
132 k
k
n
a1r
sum =52,084
sum 728
Vocabulary • An Infinite Geometric Series is a geometric
series with infinite terms.
Formula for a convergent infinite geometric series
S =
If r <1 then the _______ can be found (converges)
If r 1 then the _______ can’t be found (diverges)
)1(1
r
a
SUM
SUM≥
Examples :
1. Find the sum of the infinite geometric series 3 + 1.2 + 0.48 + 0.192 + …
a. Find the partial sum (S4)
b. Determine the ratio
2. Find the sum of the infinite geometric series 8 + 9.6 + 11.52 + 13.824 + …
a. Find the partial sum (S4)
b. Determine the ratio
Example 1:
A) Find the sum of the infinite geometric series 3 + 1.2 + 0.48 + 0.192 + …
r = .4
B) Find the sum of the infinite geometric series 8 + 9.6 + 11.52 + 13.824 + …
r= 1.2 so it is divergent
s =3
1−.4=5
Example 2:
• Find the sum of the infinite geometric series below:
11
1
3kk
Example 2:
• Find the sum of the infinite geometric series below:
• r = .3=13 s =
19
1−13
=16
Example 3:
A. Write 0.2 as a fraction in simplest form.
B. Write 0.04 as a fraction in simplest form.
Example 3:
A. Write 0.2 as a fraction in simplest form.
B. Write 0.04 as a fraction in simplest form.
2
9
4
99