which-classification-best-fits-the-sequence-given-by-an-1n-n22n-for-n-1

Question

Consider the sequence defined by an=(−1)n⋅n22n\displaystyle a_n = (-1)^n \cdot \frac{$$n^2$$}{2^n} for n≥1n \ge 1. Which classification best describes the behavior and properties of this sequence?

Accepted Answer

Alternating and bounded: The (−1)n(-1)^n factor causes the signs to flip, while the denominator 2n2^n eventually grows much faster than $$n2n^2$$, causing the terms to approach zero.

Answer

Arithmetic and unbounded: The $$n2n^2 in $$the numerator causes the sequence to grow quadratically without limit.

Answer

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

Answer

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

Consider the sequence defined by an=(−1)n⋅n22n\(\displaystyle\) a_n = (-1)^n \(\cdot\) \(\frac{n^2}{2^n}\) for n≥1n \(\ge\) 1. Which classification best describes the behavior and properties of this sequence?

Consider the sequence defined by $a_{n} = (- 1)^{n} \cdot \frac{n^{2}}{2^{n}}$ for $n is greater than or equal to 1$ . Which classification best describes the behavior and properties of this sequence?