sequences eoct: may 10-11 may 10-13 and may 17-20: school starts at 7:15 for eoct testing!
TRANSCRIPT
Sequences
EOCT: May 10-11
May 10-13 and May 17-20:
School starts at 7:15 for EOCT testing!
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
• Infinite Sequence: sequence that continues infinitely – Ex: 2, 4, 6, 8, …
• Finite Sequence: sequence that ends– Ex: 2, 4, 6
• Explicit Formula: defines the nth term of a sequence.
Example 1:
A) Write the first six terms of the sequence defined by an = 4n + 5
B) Write the first six terms of the sequence defined by an = 2n2 – 1
Vocabulary
• Recursive Formula: – Uses one or more previous terms to generate
the next term.
an-1
Example 2:
A) Write the first six terms of the sequence
where a1 = -2 and an = 2an-1 – 1
B) Write the first six terms of the sequence where a1 = 4 and an = 3an-1 + 5
Arithmetic Sequences
EOCT: May 10-11
May 10-13 and May 17-20:
School starts at 7:15 for EOCT testing!
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.
• What is d? a2 – a1
– 3, 7, 11, 15, … d = 4– 8, 2, -4, -10, … d = -6
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 1:
A) Find the 10th term of a1 = 7 and an = an-1 + 6
B) Find the 7th term of a1 = 2.5 and an = an-1 - 3
d
Example 2:
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
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
Geometric Sequences
EOCT: May 10-11
May 10-13 and May 17-20:
School starts at 7:15 for EOCT testing!
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
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
Example 1
• Find the 5th term of a1 = 8 and an = 3an-1
• Find the 7th term of a1 = 5 and an = 2an-1
Example 2:
A) Find a10 of the geometric sequence 12, 18, 27, 40.5, …
B) Find a7 of the geometric sequence where a1 = 6 and r = 4
Homework
P.140 #1-16 P.145 #1-17
***Keep reviewing for your EOCT***(May 10-11)
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.
Series(M2)
EOCT: May 10-11
May 10-13 and May 17-20:
School starts at 7:15 for EOCT testing!
Series• Series: the sum of a sequence
– Sequence: 1, 2, 3, 4– Series: 1 + 2 + 3 + 4
• Summation Notation:
4
1
12n
n
Summation Notation - __________________ EX. (for the above series)
4
1
12n
n
= _______ + _______ + _______ + _______
= ____ + _____ + _____ + _____ = _____
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
Not in packet!!
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
Extra Example:(Not in packet)
• Evaluate 5
2
1
(2 3 2)m
m m
Homework:
P.135 #18-24
*work on Benchmark Practice WS*
Arithmetic Series(M2)
EOCT: May 10-11
Sequences and Series Test: May 18
May 10-13 and May 17-20:
School starts at 7:15 for EOCT testing!
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 2:
A) Given 3 + 12 + 21 + 30 + …, find S25
B) Given 16, 12, 8, 4, …, find S11
Example 3:
A) Evaluate
B) Evaluate
12
1(6 2 )
kk
21
1(5 4 )
kk
Geometric Series (M2)
EOCT: May 10-11
Sequences and Series Test: May 18
May 10-13 and May 17-20:
School starts at 7:15 for EOCT testing!
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 S10 to the nearest tenth.
Example 2:
• Evaluate
• Evaluate
17
14( 5)k
k
16
132 k
k
n
a1r
Homework
• P. 141 #16-27
• P. 145 #18-23
• Study/Review for EOCT!
(Sequences and Series ARE ON the EOCT)
Infinite Geometric Series (M2)
May 10-13 and May 17-20:
School starts at 7:15 for EOCT testing!
Sequences and Series Test: May 18Finals:
1st Period – May 212nd Period – May 246th Period – May 26
Vocabulary • An Infinite Geometric Series is a geometric
series with infinite terms.
Formula for infinite geometric series
S =
If r <1 then the _______ can be found
If r >1 then the _______ cannot be found
)1(1
r
a
SUM
SUM
Example 1:
A) Find the sum of the infinite geometric series 3 + 1.2 + 0.48 + 0.192 + …
B) Find the sum of the infinite geometric series 8 + 9.6 + 11.52 + 13.824 + …
Example 2:
• Find the sum of the infinite geometric series below:
11
1
3kk
Example 3: NOT IN PACKET
A. Write 0.2 as a fraction in simplest form.
B. Write 0.04 as a fraction in simplest form.
Homework
• P. 147 #32 – 45 (M2 – Purple)