Expressing Sequences Explicitly
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Transcript of Expressing Sequences Explicitly
Expressing Sequences ExplicitlyBy: Matt Connor
Fall 2013
•Pure Math
•Analysis
•Calculus and Real Analysis
•Sequences
• Sequence- A list of numbers or objects in a specific order
• 1,3,5,7,9,.....
• Finite Sequence- contains a finite number of terms
• 2,4,6,8
• Infinite Sequence- contains an infinite number of terms
• 2,4,8,16, ........
•Arithmetic Sequence- add or subtract a constant to get from one term to the next
•88, 77, 66, 55,.......
•Geometric Sequence- multiply or divide by a common ratio to get from one term to the next
•6, 12, 24, 48,........
•Recursive Formula- formula for a sequence that relates the previous term(s) to find the new one.
•ex: An = A(n-1)+ 4
•Explicit Formula- formula that finds any term in the sequence without knowing any other terms.
•ex: An = 1+ 2(n-1)
•all you need to know is n
• General Forms• Recursive formula
• An = A(n-1) + d• Explicit formula
• An = A1 + d(n-1)
Arithmetic Sequences
Geometric Sequences
• General Forms• Recursive formula
• An = r(An-1)• Explicit formula
• An = A1 (rn-1)
•What about sequences that are not arithmetic or geometric?
•This means they do not have a common constant or ratio
•These are commonly called Fibonacci-type
•The difficult thing about these is finding an explicit formula
• Now we will go through deriving an explicit formula for the Fibonacci Sequence
• We know the relational formula is • An = An−1 + An−2
• We guess an explicit formula of the form An =Cxn and plug it in to the relational equation and get
• Cxn = Cxn−1 + Cxn−2
Fibonacci Sequence Explicit Formula
•Cxn = Cxn−1 + Cxn−2 this will always simplify to an equation with the same coefficients as the relational equation,
•x2 = x + 1
•Then we collect the terms on one side to use the quadratic formula.
•x2 −x−1=0
•The quadratic formula gives us x=(1/2)(1±√5)
•Therefore: An= B((1/2)(1+√5))n + C((1/2)(1-√5))n
•Next we use the first two Fibonacci numbers to find two equations representing B and C
•A0=1 and A1=1
•This gives us two equations for B and C
•B+C=1 and
•B(1/2)(1+√5) + C(1/2)(1-√5)=1
•Then we simplify the second equation we have
•(B + C) + (B - C)√5 = 2 and since our first equation tells us that B+C=1 we can replace that.
•1 + (B-C)√5 = 2
•We then further simplify this to get the second of our two equations
•B+C=1 and B-C=1/√5
•If we add these two equations and simplify we can then solve for B
•B= (√5+1)/(2√5)
•And then insert the value of B to find the value of C
•C=(√5-1)/(2√5)
•One More Step!!
•If we replace the B and C in our equation for An
•This is Binet’s formula, an explicit formula for finding the nth Fibonacci number.
An=
•As you have seen finding an explicit formula for the nth term in a Fibonacci-type sequence is much more difficult.
•. . . . . but they are possible to find!
•Resources• http://www.kenston.k12.oh.us/khs/academics/m
ath/AA_11-3A_geometric_sequences_explicit.pdf
• http://www.kenston.k12.oh.us/khs/academics/math/AA_11-2A_arithmetic_sequences.pdf
• http://www.geom.uiuc.edu/~demo5337/s97b/fibonacci.html
• http://faculty.mansfield.edu/hiseri/MA1115/1115L30.pdf