6. If the arithmetic mean of the focal distance and the length of the major axis of an ellipse equals the geometric mean of the length of the major axis and the length of the minor axis, then the value of the eccentricity of the ellipse should be in the interval ( ).
(A) $\left(0, \frac{1}{2}\right)$
(B) $\left(\frac{1}{2}, \frac{2}{3}\right)$
(C) $\left(\frac{2}{3}, \frac{3}{4}\right)$
(C) $\left(\frac{3}{4}, 1\right)$
6.C.

According to the problem, we have
$$
\begin{array}{l}
a+c=2 \sqrt{a b} \\
\Rightarrow 1+e=2 \sqrt{\frac{b}{a}}=2 \sqrt[4]{\frac{a^{2}-c^{2}}{a^{2}}}=2 \sqrt[4]{1-e^{2}} \\
\Rightarrow(1+e)^{4}=16\left(1-e^{2}\right) \\
\Rightarrow(1+e)^{3}=16(1-e) \\
\Rightarrow(1+e)^{3}+16(e-1)=0 . \\
\text { Let } f(x)=(x+1)^{3}+16(x-1) \\
=(x+1)^{3}+16(x+1)-32 .
\end{array}
$$

It is easy to see that $f(x)$ is an increasing function on $\mathbf{R}$.
And $f\left(\frac{2}{3}\right)0$, so the equation $f(x)=0$ has a unique real root in $\left(\frac{2}{3}, \frac{3}{4}\right)$.