Which of the following statements is an example of a proposition that could be proven using mathematical induction?
Question 2
When proving a statement $P(n)$ by mathematical induction, what is the significance of the base case?
Question 3
Suppose you are proving that the sum of the first $n$ terms of an arithmetic progression with first term $a_1$ and common difference $d$ is given by $S_n = \frac{n}{2}(2a_1 + (n-1)d)$. What would be the inductive hypothesis for this proof?
Question 4
Consider the statement: For all integers $n \ge 1$, $4^n - 1$ is divisible by $3$. If we assume $P(k): 4^k - 1$ is divisible by $3$ for some integer $k \ge 1$, which of the following expressions represents $P(k+1)$?
Question 5
In a proof by mathematical induction, after establishing the base case, what is the primary objective of the inductive step?
