Mathematical Induction

First prove (n+1)ⁿ⁻¹ < nⁿ for integer n ≥ 2. When n = 2 ,
3¹ < 2² is true. Assuming that (k+1)ᵏ⁻¹ < kᵏ is true ,
i.e. kᵏ / (k+1)ᵏ⁻¹ > 1 .When n = k+1 , (k+1)ᵏ⁺¹ / (k+2)ᵏ
= (k+1)ᵏ⁻¹ / (k+2)ᵏ⁻¹ * (k+1)² / (k+2)
> kᵏ⁻¹ / (k+1)ᵏ⁻¹ * (k+1)² / (k+2) .............. Tips : (k+1)/(k+2) > k/(k+1)
> kᵏ⁻¹ / (k+1)ᵏ⁻¹ * k
= kᵏ / (k+1)ᵏ⁻¹
> 1i.e. (k+2)ᵏ < (k+1)ᵏ⁺¹ . ∴ The proposition is true for n = k+1.By mathematical induction the proposition is true for for all integer n ≥ 2.
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nⁿ < (n!)² for all integer n > 2. When n = 3 ,
3³ < (3!)² i.e. 27 < 36 is true.Assuming that kᵏ < (k!)² for k >2 ,
when n = k+1 ,
(k+1)ᵏ⁺¹ = (k+1)ᵏ⁻¹ (k+1)² < kᵏ (k+1)² < (k!)² (k+1)² = ((k+1)!)²∴ The proposition is true for n = k+1.By mathematical induction the proposition is true for for all integer n > 2.