The same procedure applies to an inequality P(n). Use the assumed inequality to connect to the next step.
Example ① Prove 2ⁿ ≧ n+1 (n≧1).
Solution:n=1: 2≧2 OK.
Assume n=k: 2ᵏ ≧ k+1. For n=k+1: 2ᵏ⁺¹ = 2·2ᵏ ≧ 2(k+1) = 2k+2 ≧ k+2.
Therefore, 2ⁿ ≧ n+1.
Solution:n=1: 2≧2 OK.
Assume n=k: 2ᵏ ≧ k+1. For n=k+1: 2ᵏ⁺¹ = 2·2ᵏ ≧ 2(k+1) = 2k+2 ≧ k+2.
Therefore, 2ⁿ ≧ n+1.
Key Point
- For inequalities, pay attention to strengthening the result in the correct direction using the assumption
Practice Problems
- Give an outline for proving 2ⁿ > n² for n≧4.
Solutions and Explanations
- Solution:n=4: 16>16 → false. In fact, begin with n=5, where 32>25. Assume 2ᵏ>k². For n=k+1: 2ᵏ⁺¹=2·2ᵏ>2k². Since 2k²≧(k+1)² for k≧3, the result follows.