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Unformatted text preview: e sequence given by the initial
condition a0=2 and the recurrence an = an1 + 3, we
get the closed formula an=3n+2. 12 Fibonacci Sequence
Recall that the Fibonacci sequence is given by the
initial conditions f0=0 and f1=1 and the recurrence
relation fn = fn1+fn2. A closed formula is given by
1
fn = √
5 √ n
√ n
1+ 5
1− 5
−
2
2 13 Remark There are many techniques available to solve
recurrence relations. We will study some of them
in depth later, including methods that allow us to
derive the closed form solution to the Fibonacci
sequence. 14 Summations 15 Sums
We use the notations
n
k=m ak = am + am+1 + · · · + an The letter k is called the index of summation. 16 Example 5
2 2 2 3 2 k =1 +2 +3 +4 +5 k=1 = 1 + 4 + 9 + 16 + 25
= 55 17 2 Remark
When counting the number of operations in the
analysis of an algorithm, we get sums when
counting the number of operations within a for
loop.
The notation becomes particularly useful when
counting operations in nested loops...
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This note was uploaded on 03/24/2014 for the course CSCE 222 taught by Professor Math during the Fall '11 term at Texas A&M.
 Fall '11
 math

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