Task A-2.1. (8 points)

Simplify the fraction $\frac{\left(2^{n-1}+1\right)^{2}-4^{n-1}}{4^{n}+2^{n+1}+1}$.
## Solution.

$$
\frac{\left(2^{n-1}+1\right)^{2}-4^{n-1}}{4^{n}+2^{n+1}+1}=\frac{4^{n-1}+2 \cdot 2^{n-1}+1-4^{n-1}}{4^{n}+2 \cdot 2^{n}+1}=\frac{2^{n}+1}{\left(2^{n}+1\right)^{2}}=\frac{1}{2^{n}+1}
$$

For 4 points: good method with a minor error.

For 8 points: correct result and method.