Prove that for all n greaterthanorequalto 4 the inequality 2

Prove that for all n greaterthanorequalto 4 the inequality 2^n

Solution

we will going to prove it by induction,

the steps to n =4,and in this case certainly we have,
4! > 2 to power 4(24>16).
next the induction step,
assume that the inequality holds for some value of n>= 4,

hence we assume n!>2 to power n,and look at what happenes for n+1,

(n+1)! = n!(n+1) > 2n (n+1) > 2n .2 = 2n+1

hence,the inequality also holds for n+1.consequently it holds for every n>=4,

where 2n = 2 to the power n and 2n+1 means 2 to the power n+1