Question

Prove that 2^n > 1+n√2^n-1, for every n>2, using linear inequalities

Answer 1

$$2^{(n-1)/2}> 1.4$$

when n > 2

$$2^{(n+1)/2} > 2.8$$ $$(2^{(n+1)/2} - n)\uparrow$$

if n grows

So the less n, the less expression . So if n = 2 - expression is equal 0.8. If n > 2 then:

$$2^{(n+1)/2} - n> 0.8$$

$$2^{(n-1)/2}*(2^{(n+1)/2} - n)> 1.4*0.8 = 1.12 > 1$$

So

$$2^{(n-1)/2}*(2^{(n+1)/2} - n) - 1 >0$$

$$2^n> 1+n*{\sqrt2}^{n-1}$$