Please simplify all formulas

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Chapter 11

• Please simplify all formulas

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A sequence is a function whose domain is the set
of consecutive integers, if not specified, the
domain starts with 1. The numbers are called
terms. Finite sequences end, infinite sequences
go forever.
Find the first 6 terms of this sequence
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Writing rules, some guidelines 1) If it goes up
by the same number, its probably arithmetic
(linear). 2) If it seems to be multiplied by the
same number, its probably geometric
(exponential). 3) If it changes signs, its
either (-1)n (if it starts negative) or (-1)n1
if it starts positive. 4) If it seems to
multiply by an increasing amount, it may be of
some form (n(n1)) or something of that
nature. 5) Sometimes it helps to do the
numerator and denominator separately.
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Find the next term and write the rule.
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Graphing Sequences
n 1
a1 1
a2 3
n 2
n 3
a3 5
n 4
a4 7
a5 9
n 5
2
1
1
2
6
n 1
a1 -3
a2 0
n 2
n 3
a3 5
2
n 4
a4 12
-2
1
2
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SUMMATION (SIGMA) NOTATION
SUM (OR ADD UP) ALL OF THE TERMS IN THE
SEQUENCE When terms of a sequence are added, its
called a SERIES!
This is the UPPER LIMIT of SUMMATION (Where you
stop)
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This is the LOWER LIMIT of SUMMATION (Where you
start)
This is the INDEX of SUMMATION (Tells you where
to plug in)
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Writing a series.
2 4 6 8 10 12 20

• Find pattern
• Find Formula
• (use i for variable)
• Find limits
• 4) Write Formula

Add by 2
2i
2i 20
2i 2
i 1
i 10
1
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Writing a series.
1 2 3 4 5 6

• Find pattern
• Find Formula
• (use i for variable)
• Find limits
• 4) Write Formula

Numbers are up by 1, sign changes.
(-1)i-1 (i)
ALTERNATOR Its a sign changer.
means infinity
(-1)i-1 (i) 1
i 1
1
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Writing a series.
1 4 9 16 25

• Find pattern
• Find Formula
• (use i for variable)
• Find limits
• 4) Write Formula

Numbers are squared, sign changes.
(-1)i (i2)
means infinity
(-1)i (i2) -1
i 1
1
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You can factor out a NUMBER when possible.
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Sometimes, you add it one at a time, sometimes
you use the formula.
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11.2 Arithmetic Sequences and Series
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What do I add or multiply by to get the next
number?
Add its an arithmetic sequence.
Multiply its a geometric sequence
Add 4
Arithmetic
Times 2
Geometric
Add 1
Arithmetic
Add 3
Arithmetic
Times 1\3
Geometric
Times 2
Geometric
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Common Difference, how much each number changes
by. We use the variable d.
Common Ratio how much each number multiplies
by, we use the variable r.
Add 4
d 4
Times 2
r 2
Add 1
d 1
Add 3
d -3
Times 1\3
r 1\3
Times 2
r 2
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Is it arithmetic? What is the common
difference? How do you make a formula to find the
nth term? (difficult and easy versions) What is
the sum of a certain number of terms.
Is it geometric? What is the common ratio? How do
you make a formula to find the nth term? What is
the sum of a certain number of terms. How do you
find the sum of an infinite number of terms.
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Formula for finding an is an a1 (n-1)d
nth term
Term you want
Common Difference
First term
Write the rule for the nth term of this sequence.
5, 8, 11, 14,

• Find a1
• 2) Find d
• 3) Write
• Formula

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• Find a1
• 2) Find d
• 3) Write
• Formula

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an 5 (n-1)3
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Write the rule for the nth term of this sequence,
then find a10.

• Find a1
• 2) Find d
• 3) Write
• Formula
• 4) a10

• Find a1
• 2) Find d
• 3) Write
• Formula
• 4) a10

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Write the rule for the nth term of this sequence
n
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Find the sum of these series.

• Write formula
• Find a1
• Find an
• Find n
• 5) Solve

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Find n for the given sum Sn
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In geometric sequences, the ratio of any term to
the previous term is constant, which is called
the common ratio and represented by r. Basic
Rules
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Nice thing is we dont need the last term.
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WATCH PEMDAS!!!
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All the rules weve used so far have been
explicit rules. Now we will use and make
recursive rules. Recursive rules give the
beginning term or terms and give a recursive
equation to show how they relate.
a1 a2 a3 a4
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Write the explicit and recursive rule for the
sequence
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The ones we did earlier have a special term name
a1 1 a2 2 a3 6 a4 24 a5 120 This is a
factorial sequence.
a1 1 a2 1 a3 2 a4 3 a5 5 This is a
Fibonacci Sequence, where you take the two
previous terms and add them together to get the
next term.
If they dont say its geometric or arithmetic,
be on the lookout for other weird special
patterns.