Principle of Mathematical Induction

What is the first step in the Principle of Mathematical Induction?

In induction, the step where we assume the statement is true for n = k is called:

If P(n) is the statement to prove, which of the following correctly represents the inductive step?

Using induction, prove that the sum of the first n natural numbers is \( \frac{n(n+1)}{2} \). What is the inductive hypothesis?

Which of the following statements is NOT a valid use of mathematical induction?

Prove by induction: For all n ≥ 1, \( 2^n > n^2 \) holds for which values of n?

If P(n): \( 1^3 + 2^3 + ... + n^3 = \left( \frac{n(n+1)}{2} \right)^2 \), what is the inductive step to prove?

Which of the following is a correct base case for proving the statement: "Sum of first n odd numbers is \( n^2 \)" by induction?

Using induction, prove that \( 7^n - 1 \) is divisible by 6 for all n ≥ 1. What is the inductive hypothesis?

Which of the following is a common mistake when applying mathematical induction?