2. Which of the following expressions is greater:

$$
A=\frac{1+a+a^{2}+\ldots+a^{n-1}}{1+a+a^{2}+\ldots+a^{n}} \text { or } B=\frac{1+b+b^{2}+\ldots+b^{n-1}}{1+b+b^{2}+\ldots+b^{n}} \text {, if } a>b>0 \text {. }
$$
Solution. The expression $\frac{1}{A}$ can be written in the form:

$$
\frac{1}{A}=\frac{1+a+a^{2}+\ldots+a^{n}}{1+a+a^{2}+\ldots+a^{n-1}}=1+\frac{a^{n}}{1+a+a^{2}+\ldots+a^{n-1}}=1+\frac{1}{\frac{1}{a^{n}}+\frac{1}{a^{n-1}}+\ldots+\frac{1}{a}}
$$

Similarly for $B$, the expression $\frac{1}{B}$ can be written in the form:

$$
\frac{1}{B}=1+\frac{1}{\frac{1}{b^{n}}+\frac{1}{b^{n-1}}+\ldots+\frac{1}{b}}
$$

However, $a>b>0$, so $0\frac{1}{B}$, which means $B>A$.