When proving a statement using mathematical induction, which of the following is a common error to avoid?

Steps to Solve

1. Forgetting to prove the base case

A common mistake is to skip the base case (usually $n = 0$ or $n = 1$). Even if the inductive step is correct, the entire proof is invalid if the base case is false. The base case acts as the foundation upon which the rest of the proof rests.

2. Incorrectly applying the inductive hypothesis

The inductive hypothesis assumes the statement is true for $n = k$. A mistake is not using this assumption properly or making invalid manipulations while applying it in the inductive step.

3. Errors in algebraic manipulation during the inductive step

The inductive step often involves algebraic manipulations to show that if the statement holds for $n = k$, it also holds for $n = k + 1$. Errors in algebra can lead to an incorrect conclusion.

4. Not showing the statement is true for $n = k + 1$

The objective of the inductive step is to prove that the statement holds for $n = k + 1$. A common error is to stop before actually demonstrating this. The proof must clearly show that the statement at $n = k + 1$ follows logically from the inductive hypothesis (the statement at $n = k$).

5. Assuming what needs to be proven

It is an error to assume the statement is true for $n = k + 1$ when trying to prove it. The goal is to show it is true based on the assumption that it is true for $n = k$.