21. (12 points) Let $f _ { n } ( x )$ be the sum of the terms of the geometric sequence $1 , x , x ^ { 2 } , \cdots , x ^ { n }$, where $x > 0$, $n \in \mathrm {~N} , ~ n \geq 2$. (I) Prove that the function $\mathrm { F } _ { n } ( x ) = f _ { n } ( x ) - 2$ has exactly one zero in $\left( \frac { 1 } { 2 } , 1 \right)$ (denoted as $x _ { n }$), and $x _ { n } = \frac { 1 } { 2 } + \frac { 1 } { 2 } x _ { n } ^ { n + 1 }$; (II) Consider an arithmetic sequence with the same first term, last term, and number of terms as the above geometric sequence, with sum $g _ { n } ( x )$. Compare the sizes of $f _ { n } ( x )$ and $g _ { n } ( x )$, and provide a proof.
Choose one of questions 22, 23, or 24 to answer. If you do more than one, only the first one will be graded. Mark the box number of your chosen question with a 2B pencil on the answer sheet.
21. (12 points) Let $f _ { n } ( x )$ be the sum of the terms of the geometric sequence $1 , x , x ^ { 2 } , \cdots , x ^ { n }$, where $x > 0$, $n \in \mathrm {~N} , ~ n \geq 2$.\\
(I) Prove that the function $\mathrm { F } _ { n } ( x ) = f _ { n } ( x ) - 2$ has exactly one zero in $\left( \frac { 1 } { 2 } , 1 \right)$ (denoted as $x _ { n }$), and $x _ { n } = \frac { 1 } { 2 } + \frac { 1 } { 2 } x _ { n } ^ { n + 1 }$;\\
(II) Consider an arithmetic sequence with the same first term, last term, and number of terms as the above geometric sequence, with sum $g _ { n } ( x )$. Compare the sizes of $f _ { n } ( x )$ and $g _ { n } ( x )$, and provide a proof.
\section*{Choose one of questions 22, 23, or 24 to answer. If you do more than one, only the first one will be graded. Mark the box number of your chosen question with a 2B pencil on the answer sheet.}