5. Given that $a, b$ are unequal positive numbers, insert two sets of numbers $x_{1}, x_{2}, \cdots, x_{n}$ and $y_{1}, y_{2}, \cdots, y_{n}$ between $a$ and $b$, such that $a, x_{1}, x_{2}, \cdots, x_{n}, b$ form an arithmetic sequence, and $a, y_{1}, y_{2}, \cdots, y_{n}, b$ form a geometric sequence, then which of the following inequalities (1) $\frac{1}{n}\left(x_{1}+x_{2}+\cdots+x_{n}\right)>\sqrt{a b}+ \left(\frac{\sqrt{a}-\sqrt{b}}{2}\right)^{2}$
(2) $\frac{1}{n}\left(x_{1}+x_{2}+\cdots+x_{n}\right)>\frac{a+b}{2}$;
(3) $\sqrt[n]{y_{1} y_{2} \cdots y_{n}}< \sqrt{a b}$; (4) $\sqrt[n]{y_{1} y_{2} \cdots y_{n}}<\frac{a+b}{2}-\left(\frac{\sqrt{a}-\sqrt{b}}{2}\right)^{2}$ are true? 
A. (1) (3)
B. $(1)(4)$
C. (2) $(3)$
D. $(2)(4)$
5. B.
$$
\begin{array}{l}
x_{1}+x_{n}=x_{2}+x_{n-1}=\cdots=a+b, \text{ then } \frac{1}{n}\left(x_{1}+x_{2}+\cdots+x_{n}\right)=\frac{1}{2 n}\left[\left(x_{1}+x_{n}\right)+\left(x_{2}+\right. \right. \\
\left.\left.x_{n-1}\right)+\cdots+\left(x_{n}+x_{1}\right)\right]=\frac{1}{2 n} \cdot n(a+b)=\frac{a+b}{2}=\sqrt{a b}+2\left(\frac{\sqrt{a}-\sqrt{b}}{2}\right)^{2}>\sqrt{a b}+\left(\frac{\sqrt{a}-\sqrt{b}}{2}\right)^{2} . \\
\text{ Also, } y_{1} y_{n}=y_{2} y_{n-1}=\cdots=a b, \text{ then } \sqrt[n]{y_{1} y_{2} \cdots y_{n}}=\sqrt[2 n]{\left(y_{1} y_{n}\right)\left(y_{2} y_{n-1}\right) \cdots\left(y_{n} y_{1}\right)}=\sqrt[2 n]{(a b)^{n}}= \\
\sqrt{a b}=\frac{a+b}{2}-2\left(\frac{\sqrt{a}-\sqrt{b}}{2}\right)^{2}<\frac{a+b}{2}-\left(\frac{\sqrt{a}-\sqrt{b}}{2}\right)^{2} .
\end{array}
$$