Question

For which positive integers n is n + 6 < (n^2 − 8n)/16? Prove your answer using mathematical induction.

Answer 1

To determine the positive integers for which the inequality holds, we simplify the inequality and use mathematical induction, starting with the base case of n=1, and then proceeding with the inductive step.

Step-by-step explanation:

To determine for which positive integers n the inequality n + 6 < (n^2 − 8n)/16 holds, we can approach the problem using mathematical induction. First, let's simplify the inequality:

1. Multiply both sides by 16 to clear the fraction: 16n + 96 < n^2 − 8n.
2. Bring all terms to one side to form a quadratic inequality: n^2 − 24n − 96 > 0.
3. Find the roots of the corresponding quadratic equation n^2 − 24n − 96 = 0 to determine the critical values.

By factoring or using the quadratic formula, we can find the values of n for which the inequality is satisfied. For the base case in mathematical induction, we can start by checking if the inequality holds for the smallest possible positive integer, which is n=1. Then, for the inductive step, we assume the inequality is true for some k and prove it for k + 1. This method will help us to establish the range of n satisfying the inequality.

To prove whether the initial inequality holds through mathematical induction, a separate, detailed step-by-step proof would be required.