Infinitely many concentric circles are drawn, such that the ratio of the radius of each circle to the radius of the next larger circle is $k$, where $0<k<1$. The regions between consecutive circles are shaded alternately, starting with the outermost ring shaded. (The innermost disk is not part of any ring.) The shaded area is $\frac{25}{34}$ of the area of the largest disk. What is $k$?

1. Let the largest circle have radius $1$. The disks have areas $\pi,\ \pi k^2,\ \pi k^4,\dots$, so the shaded rings have areas $\pi(1-k^2),\ \pi(k^4-k^6),\ \pi(k^8-k^{10}),\dots$.
2. This is a geometric series with first term $\pi(1-k^2)$ and ratio $k^4$, so the shaded area is $\dfrac{\pi(1-k^2)}{1-k^4}=\dfrac{\pi}{1+k^2}$.
3. Hence $\dfrac1{1+k^2}=\dfrac{25}{34}$, so $k^2=\dfrac9{25}$ and $k=\boxed{\dfrac35}$.