<p> UBE 501 DISCRETE MATHEMATICS Fall 2011</p><p>Problem Set #7</p><p>1)Show that the sequence{ n } is a solution of the recurrence relation  n = 3 n1 +4 n2 if</p><p> a)  n = 0</p><p> b)  n =1 n c)  n =(-4) n d)  n =2(-4) +3</p><p>2)Find the solution to each of these recurrence relations with the given initial conditions. Use telescoping approach.</p><p> a)  n =   n1 , 0  5</p><p> b)  n = n1  3, 0  1-</p><p> c)  n = n1  n, 0  4</p><p> d)  n = 2 n1  3, 0  1</p><p> e)  n =(n 1) n1 , 0  2</p><p> f)  n = 2n n1 , 0  3</p><p> g)  n =  n1  n 1, 0  7</p><p>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?</p><p>  2    2    6 4)Find the solution to n n1 n2 n3 for n =3, 4, 5,…, with 0 =3, 1 and</p><p> 2  0</p><p>2 5) a)Find the solutions of the recurrence relation  n = 2 n1  2n .</p><p> b)Find the solution of the recurrence relation in part(a) with initial condition 1 = 4.</p><p>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.</p><p>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 .]</p>