Question

prove that 2^n > 1+n*√2^(n-1) for all n>2

Answer 1

Prove that 2ⁿ > 1+n*√2^n-1 for all n>2

Solution.

2ⁿ-1 > n*√2^n-1 square both sides of inequality

(2ⁿ-1)² > n²*2^n-1

2²ⁿ-2*2ⁿ+1 > n²*2^n-1 move n²*2^n-1 to the left side, than take 2ⁿ out of brackets

2ⁿ *( 2ⁿ - 2 – n²/2) +1 > 0

Obviously 2ⁿ > 0

So now we should just prove that ( 2ⁿ - 2 - n²/2) >= 0

2ⁿ - 2 >= n²/2

2(2ⁿ - 2) >= n²

As function graphs show

(where red graph – n²; blue graph - 2(2ⁿ - 2) and green one – n=2)

When n > 2 blue graph is higher than red one. That means that 2(2ⁿ - 2) >= n2.

And that proves that 2ⁿ > 1+n*√2^n-1 for all n>2