Exercise 3, p. 118

Function to return the value of for x,

1. , so this is valid.
2. Assume Q(k), and consider Q(k+1):

   

   using the case. So we have verified the loop invariant by induction.

3. At termination: , hence .

Exercise 8, p. 118

so gcd(1018215,2695)=55, by the Euclidean algorithm.

Exercise 13, p. 120

Note in particular that i is stepping down in this case by 1, rather than up as it has in the other examples.

1. , so this is valid.
2. Assume Q(k), and consider Q(k+1):

   

   using the case. So we have verified the loop invariant by induction.

3. At termination: , so .