### 5. Consider the following program segment: ```text i := 1 total := 1 while i < n do i := i + 1 total := total + i ```` Let ( p ) be the proposition: [ p:\quad ( \text{total} = \frac{i(i+1)}{2} \ \text{and}\ i \le n ) ] Use mathematical induction to prove ( p ) is a loop invariant. --- ## Solution ### Basis Step Before the loop is entered, ( p ) is true since [ \text{total} = 1 = \frac{1(1+1)}{2} ] and [ i \le n ] --- ### Inductive Step Suppose ( p ) is true and ( i = k < n ) after the ( k - 1 )-th execution of the loop. Since ( p ) is true: [ \text{total} = \frac{k(k+1)}{2} ] Suppose that the while loop is executed again. Then ( i ) is incremented to [ i = k + 1 ] Total becomes: [ \begin{aligned} \text{total} &= \text{total}_{\text{prev}} + i \ &= \frac{k(k+1)}{2} + (k+1) \quad \text{(by inductive hypothesis)} \ &= \frac{k(k+1) + 2(k+1)}{2} \ &= \frac{(k+1)(k+2)}{2} \end{aligned} ] --- After the loop, ( i \le n ) and `total` is still of the required form. Therefore, ( p ) is a loop invariant. ∎ 使用中文解答