Prove that the product $\prod_{i=1}^n \left(1-(1/2)^i\right)\ge\left (1/4 + 1/2^{n+1}\right)$ for any integer $n$

Question

Expanded out this would be $(1-(1/2))(1-(1/4))(1-(1/8)) \cdots (1-(1/2^{n})) \ge (1/4 + 1/2^{n+1})$. I'm currently working to solve this problem but I cannot come to a reasonable conclusion. I am trying to solve this using induction by proving it for $n=k$ and $n=k+1$ but I cannot find a answer that is a reasonable proof.

Answer 1

HINT:

$$\left(\frac14 +\left(\frac12 \right)^{n+1}\right)\left(1-\left(\frac12\right)^{n+1}\right)=\frac14+\left(\frac12\right)^{n+2}\left(\frac32 -\left(\frac12\right)^n\right)$$

Can you finish now?