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Algorithms Homework 2

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Homework 2

1. Prove that sum = i
2 after every iteration of the for loop below:
Input: n: nonnegative integer
Output: n
2
1 sum = 0
2 for i = 1 to n do
3 sum = sum + 2i − 1
4 end
5 return sum
2. Use strong induction on n to prove that the algorithm below computes 2n
for all n ≥ 0.
Input: n: nonnegative integer
Output: 2
n
1 Algorithm: QuickPow
2 if n = 0 then
3 return 1
4 else if n is even then
5 t = QuickPow(n/2)
6 return t
2
7 else
8 t = QuickPow((n − 1)/2)
9 return 2t
2
10 end
1

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