summation

# An arithmetic function is a discrete analogue of a

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Unformatted text preview: metic function is a discrete analogue of a linear function dx+a. 6 Strings The data type of a string is nothing but a sequence of finite length. 7 Recurrence Relations 8 Recurrence Relation A recurrence relation for a sequence {an} expresses the term an in terms of previous terms of the sequence. The initial conditions for a recursively defined sequence specify the terms before the recurrence relation takes effect. A sequence is called a solution of a recurrence relation if its terms satisfy the recurrence relation. 9 Example Let {an} be the sequence defined by the initial condition a0=2 and the recurrence relation an = an-1 + 3 Then a1 =2+3=5, a2=5+3=8, a3=8+3=11,... 10 Fibonacci Sequence Recall that the Fibonacci sequence is defined by the initial conditions: f0=0 and f1=1 and the recurrence relation fn = fn-1+fn-2 for n >= 2. Hence, {fn} = {0,1,1,2,3,5,8,13,...} 11 Solving Recurrence Relations We say that we have solved a recurrence relation if we can find an explicit formula, called a closed formula, for the terms of the sequence. Example: For th...
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