Consider the sequence defined by $a_n=(-1{)}^n\cdot \frac{n^2}{2^n}$ for $n\ge 1$. Which classification best describes the behavior and properties of this sequence?

Alternating and bounded: The $(-1{)}^n$ factor causes the signs to flip, while the denominator $2^n$ eventually grows much faster than $n^2$, causing the terms to approach zero.

Arithmetic and unbounded: The $n^2$ in the numerator causes the sequence to grow quadratically without limit.

Geometric: The formula represents a geometric sequence because it contains a constant base raised to the power of $n$.

Monotonic increasing: The terms always increase in value as $n$ becomes larger.