1)Show That the Sequence Is a Solution of the Recurrence Relation If
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UBE 501 DISCRETE MATHEMATICS Fall 2011
Problem Set #7
1)Show that the sequence{ n } is a solution of the recurrence relation n = 3 n1 +4 n2 if
a) n = 0
b) n =1 n c) n =(-4) n d) n =2(-4) +3
2)Find the solution to each of these recurrence relations with the given initial conditions. Use telescoping approach.
a) n = n1 , 0 5
b) n = n1 3, 0 1-
c) n = n1 n, 0 4
d) n = 2 n1 3, 0 1
e) n =(n 1) n1 , 0 2
f) n = 2n n1 , 0 3
g) n = n1 n 1, 0 7
3) a)Find a recurrence relation fort he number of bit strings of lenght n that contain three consecutive 0s. b)What are the initial conditions? c)How many bit strings of length seven contain three consecutive 0s?
2 2 6 4)Find the solution to n n1 n2 n3 for n =3, 4, 5,…, with 0 =3, 1 and
2 0
2 5) a)Find the solutions of the recurrence relation n = 2 n1 2n .
b)Find the solution of the recurrence relation in part(a) with initial condition 1 = 4.
6)Find f (n) when n =3 k , where f satisfies the recurrence relation f (n)=2 f (n/3)+4 with f (1)=1.
7)Suppose that the function f satisfies the recurrence relation f (n) 2 f ( n) log n whenever n is a perfect square greater than 1 and f (2) 1. a)Find f (16). b)Find a big-O estimate for f (n).[Hint: Maket the substitution m log n .]