Answer
$\frac{f(n+2)}{f(n)}= \frac{1}{2}$
Work Step by Step
First write the function in standard form.
\begin{equation}
\begin{aligned}
f(n)&=1000\cdot 2^{-\frac{1}{4}-\frac{n}{2}}\\
&=1000\cdot 2^{-\frac{1}{4}}\cdot 2^{-\frac{n}{2}}\\
&= \frac{1000}{2^{\frac{1}{4}}}\cdot\left( \frac{1}{2^{\frac{1}{2}}}\right)^n\\
&= \frac{1000}{\sqrt{\sqrt{2}}}\cdot\left( \frac{1}{\sqrt{2}}\right)^n
\end{aligned}
\end{equation}
Let.
\begin{equation}
\begin{aligned}
b&=\frac{1}{\sqrt{2}}\\
&= \frac{1}{b_0}\\
a&=\frac{1000}{\sqrt{\sqrt{2}}}\\
&= \frac{1000}{\sqrt{b_0}}
\end{aligned}
\end{equation}
Then
\begin{equation}
\begin{aligned}
f(n)&=a\cdot b^n\\
f(n+2)&= a\cdot b^{n+2}\\
&=ab^2\cdot b^n\\
\frac{f(n+2)}{f(n)}&= \frac{ab^2\cdot b^n}{a\cdot b^n}\\
&= b^2\\
&= \left( \frac{1}{\sqrt{2}} \right)^2\\
\frac{f(n+2)}{f(n)}&= \frac{1}{2}\\
\end{aligned}
\end{equation}
This means that an $A_{n}$ paper is two times the size of $A_{n+2}$. For example, $A_1$ paper is equal to two times the size of $A_3$ paper.
