Mathematical Induction Math Example 2

Solution

1. 1
   Base case ($n = 4$): $4! = 24$ and $2^4 = 16$. Since $24 > 16$, the base case holds.
2. 2
   Inductive hypothesis: Assume $k! > 2^k$ for some $k \ge 4$.
3. 3
   Inductive step: $(k+1)! = (k+1) \cdot k! > (k+1) \cdot 2^k$. Since $k \ge 4$, we have $k+1 \ge 5 > 2$.
4. 4
   Therefore $(k+1)! > 2 \cdot 2^k = 2^{k+1}$. The inequality holds for $k+1$.