Prove by induction that ?n=3 : n^2+1=3n

Question

Prove by induction that ∀n≥3 : n^2+1≥3n

Answer 1

The statement holds for n=3 since 3^2 + 1 = 10 >= 9 = 3*3

Given n^2 +1 >= 3n

(n+1)^2 + 1 = n^2 + 2n + 2 = <--- FOIL

n^2+ 2n + 1 + 1 =

(n^2+1) + (2n+1) >= <--- associative

3n +2n + 1 >= <--- by induction hypothesis

3n + 2(3) + 1 > = <--- n>=3

3n + 7 >=

3n + 3 =

3(n+1)