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1 . By definition, show that 1 k + 2 k + + nk is O ( nk + 1 ) , where k is

1. By definition,
show that 1k +2k ++ nk is O(nk+1), where k is a positive integer. (20 points)
2. Order the following functions into a list such that if f (n) comes before g(n) in the list then
f (n)= O(g(n)). If any two (or more) of the same asymptotic order, indicate which. Start with these
basic functions (10 points)
n,2n, n lg n, n3, lg n, n n3+7n5, n2+ lg n
3. Find the solution for each of the following recurrences, and then give tight bounds for T(n).(30
points)
.()=(1)+1/ with T (0)=0
.()=(1)+ with T (0)=1, where c >1 is some constant
.()=(1)+1 with T (0)=1
4. Use the master theorem to give tight asymptotic bounds for the following recurrences. (40 points)
a.()=2(
2)+n
b.()=3(
2)+
c.()=27(
3)+3
d.()=5T (
4)+ c2
e.()=2(
4)+1(Bonus 10 points)
f.()=2(
4)+n (Bonus 10 points)

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