Let a_n=(n)/(2^n). Which statement is correct?

It is decreasing, but not strictly, since a_1=a_2

It is strictly decreasing from n=1

Exercise · ZW-MONOTONE-SEQUENCE-010 · ★★★ Advanced

Let \(a_n=\frac{n}{2^n}\). Which statement is correct?

Choose one option.

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Here \(\frac{a_{n+1}}{a_n}=\frac{n+1}{2n}\le1\), with equality only for \(n=1\): \(a_1=a_2=\tfrac12\) and strictly smaller afterwards.

A real sequence \((a_n)\) is monotone when, as the index \(n\) grows, its terms either never decrease or never increase.

\[a_n\le a_{n+1}\ \ \forall n\quad\text{or}\quad a_n\ge a_{n+1}\ \ \forall n\]