FLC Ch 12 Page 1 of 18 Math 370 Precalculus Sec 12.1: Sequences What is a sequence? Notation { } or { } or Ex 1 Find each. a) b) c) if d) e) Number of ways to rearrange/permute 1, 2, and 3 symmetric group S3 is isomorphic to dihedral group D3 Note: There are two ways of counting. When order matters, we find the number of permutations (sec 12.1). When order doesn't matte r, we find the number of combinations or "choose" (sec 12.5). Defn A sequence is a function whose domain is (the natural numbers { }). Sequences are usually represented as an ordered list. The numbers in the list are called terms of the sequence. The th term is called the general term. Factorial Symbol If is an integer, the factorial symbol is defined as follows: ( ) if
Transcript
FLC Ch 12
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Math 370 Precalculus Sec 12.1: Sequences
What is a sequence?
Notation { } or { } or Ex 1 Find each.
a) b) c) if d)
e) Number of ways to rearrange/permute 1, 2, and 3 symmetric group S3 is isomorphic to dihedral group D3 Note: There are two ways of counting. When order matters, we find the number of permutations (sec 12.1). When order doesn't matte r, we find the number of combinations or "choose" (sec 12.5).
Defn A sequence is a function whose domain is (the natural numbers { }).
Sequences are usually represented as an ordered list.
The numbers in the list are called terms of the sequence. The th term is called the general term.
Factorial Symbol If is an integer, the factorial symbol is defined as follows: ( ) if
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Ex 2 Write down the first 5 terms of each sequence.
(#17) { } {
} (#19) { } {( )
}
(#21) { } {
} (#24) { } {
}
( )
is analytic everywhere (entire) Will need this in m401
Ex 3 In the following problems, the given pattern continues. Write down the th term of a sequence { } suggested by the pattern.
(#27)
(#29)
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(#31)
(#32)
(#33)
Ex 4 A sequence is defined recursively. [This means the first or first few terms are provided. The term is determined by the rule involving one or more of the previous terms.] Write down the first 5 terms. (#35) ; (#43) ; ;
(Summation) Notation
Ex 5 Write out each sum.
∑( )
∑
∑( )
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(#52)
∑( )
(#53)
∑
Note:
∑
where
from page 4
Ex 6 Express each sum using summation notation. (#59) (#60) (#62) [ ( ) ]
(#63)
( ) (
)
Ex 7 Find the sum of each sequence. (#69)
∑
(#74)
∑( )
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∑
(#76)
∑( )
Sec 12.2: Arithmetic Sequences
What is an arithmetic sequence [progression]?
∑
∑
( )( )
∑
[ ( )
]
( )
Defn An arithmetic sequence may be defined recursively as , or as
where and are real numbers. The number is the first term, and the number is called the common difference.
th Term of an Arithmetic Sequence
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Ex 8 Find the th term of the arithmetic sequence { } whose initial term and common difference are given. What is the fifty-first term? (#13)
(#18)
Ex 9 Find the indicated term in each arithmetic sequence. (#23) 90th term of
(#26) 70th term of √ √ √ Note: Arithmetic sequences are ___________________________ whose domain is restricted to . Ex 10 (#30) Give a recursive formula for the arithmetic sequence. Find a formula for the th term. 8th term is ; 18th term is
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Ex 11 Find each sum. (#35) ( ) (#37) ( )
(#46)
(#50)
∑(
)
∑[ ( ) ]
[ ( ) ]
∑[ ( ) ]
( )
Sum of First Terms of an Arithmetic Sequence
OR
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Ex 12 How many 3-digit multiples of 7 are there? Use a sequence to solve.
Sec 12.3: Geometric Sequences
What is a geometric sequence [progression]? Ex 13 Find the th term of each geometric sequence.
Ex 14 Find and the missing terms.
Defn An geometric sequence may be defined recursively as
, or as
where and are real numbers. The number is the first term, and the nonzero number is called the common ratio.
th Term of an Geometric Sequence
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Ex 15 Determine whether each given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. (#20) { } (#30) { }
(#28) {(
) }
Ex 16 (#38) Find the fifth term and the nth term of the geometric sequence where and
Ex 17 (#45) Find the indicated term of the geometric sequence. 8th term of
Ex 18 (#50) Find the nth term of the geometric sequence.
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Derive the partial sum formula for a geometric sequence. Ex 19 Find each sum.
(#56)
(#60)
(
)
∑
Sum of First Terms of a Geometric Sequence
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(#62) Use Graphing Utility
Note that
∑
∑
If the finite sum approaches a number as , we say the series ∑
converges and call
the sum of the infinite series. That is, ∑
. If the series does not converge, we say it diverges. Ex 20 Determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
(#70)
(#78)
∑ (
)
∑
Infinite Geometric Series Defn An infinite sum of the form
with first term and common
ratio , is called an infinite geometric series and is denoted by
∑
Convergence of an Infinite Geometric Series
The infinite series ∑
converges iff | | . Its sum is
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(#82)
∑ (
)
Ex 21 Show that ̅ using a series.
Extra Credit: Show ̅ using algebra.
Ex 22 (#90) Jolene wants to purchase a new home. Suppose she invests $400 per month into a mutual fund. If the per annum rate of return of the mutual fund is assumed to be 10% compounded monthly, how much will Jolene have for a down payment after the 36th deposit (3 years)? $16,712.73
( )
Amount of an Annuity An annuity is a sequence of equal periodic deposits. Suppose is the deposit in dollars made at the end of each payment period for an annuity paying percent interest per payment period. The amount of the annuity after deposits is
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Sec 12.4: Mathematical Induction
Mathematical induction is a method used to prove that statements involving natural numbers are true for all . Example in book (dominos) and "always right" example.
Ex 23 Prove that ( ) for all natural numbers . Ex 24 (#2) Prove that ( ) ( ) for all natural numbers .
Theorem The Principle of Mathematic Induction Suppose that the following two conditions are satisfied with regard to a statement about natural numbers: CONDITION I: The statement is true for the natural number 1. CONDITION II: If the statement is true for some natural number , it is also true for the next natural number . Then the statement is true for (for all natural numbers).
Important: It is extremely crucial to show each step when constructing proofs, including proofs by induction. When writing formal proofs, written statements are just as critical as algebraic ones. Proofs need to be rigorous and steps that are "obvious" should NOT be left to the reader. If it is obvious to you, show that it is. Proofs will be graded on rigor and clarity. Refer to your notes or the provided handout ("Proof by Mathematical Induction") for acceptable proofs.
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Ex 25 (#19) Prove that is divisible by 2 for all natural numbers . Ex 26 (#7) Prove that for all natural numbers .
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Ex 27 (#20) Prove that is divisible by 3 for all natural numbers .
Sec 12.5: The Binomial Theorem
There are two ways of counting -- using permutations or combinations and both count the number of ways to select distinct objects from a set of elements. When finding the permutations on the set, we take order into account. Combinations are permutations where the order does not matter. Ex 28 Evaluate each.
(#5) ( ) (#16) (
)
( ) (
) (
)
Defn " choose "
If and are integers with , the symbol ( ) is defined as (
)
( ) .
( ) may also be denoted .
B G R P
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How do we expand ( ) ? See if you notice any patterns
( ) 1 term
( ) 2 terms
( ) 3 terms
( ) 4 terms
( ) 5 terms
( ) 6 terms
Patterns: 1) # of terms: ( ) has ________ terms
2) first and last term of ( ) are: ________ and ________, respectively
3) powers of and : powers of ______________ by 1 whereas the powers of ______________ by 1
4) sum of exponents for “ ” and “ ” terms in expansion: add up to ______
What about the coefficients? The pattern can be found in Pascal’s Triangle.
Pascal’s Triangle
The Binomial Theorem Let and be real numbers. For any given natural number , we have
( ) ( ) (
) (
) (
) ∑ (
)
Note: The symbol ( ) is called a binomial coefficient.
(
)
The term containing is
(
) (
) (
)
If and are integers with , then
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Ex 29 Expand each.
(#20) ( ) ( )
(√ √ )
Ex 30 Find the indicated coefficient or term. (#34) The coefficient of in the expansion of ( ) . (#38) The sixth term in the expansion of ( ) . Subsequences
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Practice Problems 1) Convert from parametric equations to a rectangular equation. 2) Find the nth term for { }. 3) Find for .
4) Let (
)
.
a) List the first 4 terms of . b) Find using the formula. c) Does the geometric series determined by this sequence converge? If so, find the sum.
