1. **Problem Statement:** Prove using mathematical induction that for any integer $n \geq 1$, $11^n - 6$ is divisible by 5. 2. **Formula and Principle:** Mathematical induction involves two steps: - Base Case: Verify the statement for $n=1$. - Inductive Step: Assume the statement is true for $n=k$, i.e., $11^k - 6$ is divisible by 5. Then prove it is true for $n=k+1$. 3. **Base Case ($n=1$):** Calculate $11^1 - 6 = 11 - 6 = 5$. Since 5 is divisible by 5, the base case holds. 4. **Inductive Hypothesis:** Assume for some integer $k \geq 1$, $11^k - 6 = 5m$ for some integer $m$. 5. **Inductive Step:** Consider $11^{k+1} - 6$: $$ 11^{k+1} - 6 = 11 \cdot 11^k - 6 = 11(11^k) - 6 $$ Rewrite as: $$ 11^{k+1} - 6 = 11(11^k - 6) + 11 \times 6 - 6 = 11(11^k - 6) + 66 - 6 = 11(11^k - 6) + 60 $$ By the inductive hypothesis, $11^k - 6 = 5m$, so: $$ 11^{k+1} - 6 = 11 \times 5m + 60 = 5(11m + 12) $$ Since $11m + 12$ is an integer, $11^{k+1} - 6$ is divisible by 5. 6. **Conclusion:** By mathematical induction, $11^n - 6$ is divisible by 5 for all integers $n \geq 1$. **Final answer:** The statement is proven true by induction.