Quiz 4 SolutionsDilip RaghavanNovember 11, 2008Problem 1.(a) If the first slot is not a 0, then we have 3 possibilities for the first slot andan-1. If the firstslot is 0, then we nust have an odd number of 0 in what follows. The total number of sequences oflengthn-1 is 4n-1, and of thesean-1have an even number of 0s. So there are 4n-1-an-1possibilitiesfor this case. Soan=3an-1+4n-1-an-1=2an-1+4n-1.(b) The homogeneous solution isA2n. Look for a particular solution of the formp(n)=B4n. Substituting,we get the equationB4n=2B4n-1+4n-1, which simplifies to 4B=2B+1 Hence,B=12.Soan=A2n+124n. Using the initial conditiona1=3, we getA=12.Problem 2.Letanbe the number of such subsets. We find a recurrence relation for
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Natural number, consecutive integers, Dilip Raghavan, an-2 such subsets, an-1 such subsets