Question

Use induction to prove that your algorithm to compute the sum of the first n positive odd integers returns the correct value for every positive integer input.

Answer 1

Using mathematical induction, we establish a base case for n=1 and assume the algorithm's correctness for an integer k. Then, we demonstrate the algorithm's validity for k+1, ultimately proving that the algorithm correctly computes the sum of the first n positive odd integers as n² for any positive integer n.

Step-by-step explanation:

The sum of the first n positive odd integers can be proved to be correct using mathematical induction. Assume the given algorithm states that the sum of the first n odd numbers is given by the expression n². We shall prove this is true for every positive integer n.

Base Case: When n=1, the sum of the first odd number (1) is indeed 1² = 1, so the base case holds.

Induction Hypothesis: Assume the statement is true for some integer k, such that the sum of the first k odd numbers is k².

Inductive Step: We must show that the statement holds for k+1. The sum of the first k+1 odd numbers would include the sum of the first k odd numbers (which by hypothesis is k²) plus the next odd number, which is 2k+1. Therefore, the sum is k² + 2k + 1 = (k+1)², which completes the inductive step as this is precisely the sum we're aiming to prove for k+1.

Thus, by mathematical induction, we conclude that the algorithm correctly computes the sum as n² for every positive integer n.