Problem 31 - College Algebra

Question

Explain how to use mathematical induction to prove that a statement is true for every positive integer \(n\)

Step-by-Step Solution

Answer

To use mathematical induction to prove a statement is true for every positive integer, start with verifying the statement for the smallest positive integer (1 in most cases), then assume that the statement is true for some arbitrary positive integer \(k\). Lastly, prove that if it's true for \(k\), then it is also true for the next positive integer \(k+1\). If these three steps can be successfully carried out, the statement is proven via mathematical induction.

Step 1: Base Case

The base case serves as the foundation of a mathematical induction proof. Typically, it involves verifying that the statement is true for the smallest positive integer, often 1. In our case, we would show the result holds 'true' when iterating for \(n =1\)

Step 2: Inductive Hypothesis

The inductive hypothesis is the assumption that the statement is true for some positive integer \(k\). We consider that the statement holds 'true' for an arbitrary positive integer \(k\). The reason for this assumption is to apply it in the next step, known as the induction step.

Step 3: Inductive Step

In the inductive step, we must show that if the statement holds for \(k\), then it also holds for \(k+1\). This involves taking the assumption made in the inductive hypothesis and proving that the statement is true for the next integer along (i.e., \(k + 1\)). If this is achieved, then it can be said that the statement is proven for all positive integers since we have proved it for 1 (Base case) and shown that if it's true for one integer (Inductive step) it is true for the next.