Thread: Use induction to verify the inequality

The problem:
Use induction to verify the inequality
2n+1 <= 2ⁿ | n >= 3

My work thus far:
Base case n = 3:
2(3) + 1 <= 2³
7 <= 8

Inductive step:
Assume 2n+1 <= 2ⁿ is true for some n >= 3.
Prove 2(n+1) + 1 <= 2⁽ⁿ⁺¹⁾(2n+1)+2 <= 2*2ⁿ
...

I am not sure what to do from here. Replacing the 2ⁿ leaves me with 4n+2 and replacing the 2n+1 leaves me with 2ⁿ+2. I thought that with inequalities I am suppose to replace terms in the greater expression with terms from the lesser expressions so that the resulting expression equals the lesser expression or vice versa. However, with this neither replacement results in an equivalent expressions.

Originally Posted by AltF4

The problem:
Use induction to verify the inequality
2n+1 <= 2ⁿ | n >= 3

My work thus far:
Inductive step:
Assume 2n+1 <= 2ⁿ is true for some n >= 3.
Prove 2(n+1) + 1 <= 2⁽ⁿ⁺¹⁾(2n+1)+2 <= 2*2ⁿ...