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Problem 2 ( 1 2 pts . ) : Sum of natural numbers Below we give, in Java syntax, the sumn method which computes the

Problem 2(12 pts.): Sum of natural numbers
Below we give, in Java syntax, the sumn method which computes the sum of the n natural numbers
from 1 to n. For example, sumn(0)==0, sumn(1)==1, sumn(2)==3, and sumn(6)==21.
// P r e c o n d i t i o n : n >=0
int sumn ( int n ){
int i =0, t =0 ;
while ( i < n ){
i = i +1 ;
t = t + i ;
}
return t ;
}
// P o s t c o n d i t i o n : t = n *( n +1)/2
a) Find a suitable loop invariant. (2 pt.)
b) Show that the invariant holds before the loop (base case).(1 pt.)
c) Show by induction that if the invariant holds after k-th iteration, and execution takes a k+1-st
iteration, the invariant still holds (inductive step).(3 pts.)
d) Show that the loop exit condition and the loop invariant imply the postcondition t = n(n+1)/2.
(2 pts.)
e) Find a suitable decrementing function. Show that the function is non-negative before loop starts,
that it decreases at each iteration and that when it reaches 0 the loop is exited. (2 pts.)

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