Iteration:
In iteration, the loop body is repeatedly executed as long as the loop condition is true. Each time the loop body is executed, the variables are updated. However, there is also a property of the variables which remains unchanged by the execution of the loop body, This unchanging property is called the loop invariant. Loop invariant is the key to construct and to reason about iterative algorithms.

(i) Establish the loop invariant at the start of the loop.
(ii) The loop body should so update the variables as to progress toward the end and maintain the loop invariant, at the same time.
(iii) When the loop ends, the termination condition and the loop invariant should establish the input-output relation

Let P(m, n) = m + 3n. Then
P(m, n) [m, n := m + 3, n -1]
= m + 3n [m, n := m + 3, n -1]
= (m + 3) + 3(n -1)
= m + 3 + 3n - 3
= m + 3n
= P(m, n)
Since (m + 3n) [m, n : = m + 3, n - 1] = m + 3n, m + 3n an invariant of the assignment m, n := m + 3, n-1.

Recursion is another algorithm design technique, closely related to iteration. Using recursion, we solve a problem with a given input, by solving the same problem with a part of the input, and constructing a solution to the original problem from the solution to the partial input.

(i) Even though the action is the same, the state in which the action is executed is not the same. Each time we execute the action, the state changes.
(ii) Therefore, the same action is repeatedly executed, but in different states.
(iii) The state changes in such a way that the process progresses to achieve the desired input-output relation.