Rutherford's $\alpha$-particle scattering experiment eventually led to the conclusion that

The ratio of the maximum wavelengths of the Lyman and Balmer series is ....

Based on the equation $E = -2.178 \times 10^{-18} \ J \left( \frac{Z^2}{n^2} \right)$,certain conclusions are written. Which of them is not correct?

Which of the following relations is correct for the energy of the ground state of hydrogen-like species?

How many photons are required to provide $1 \, J$ of energy with a wavelength of $5000 \, \mathring{A}$?

When a metal surface is irradiated with light of frequency $x \ Hz$,the kinetic energy of emitted photoelectrons is $z \ J$. When the same metal is irradiated with light of frequency $y \ Hz$,the kinetic energy of emitted electrons is $\frac{z}{3} \ J$. What is the threshold frequency (in $Hz$) of the metal?