Which level of the single ionized carbon has the same energy as the ground state energy of hydrogen atom?

The energy of the $$n^{th}$$ level of a hydrogen-like atom with atomic number $$Z$$ is $$E_n = -\frac{13.6\, Z^2}{n^2}$$ eV. The ground state energy of hydrogen ($$Z = 1, n = 1$$) is $$E_1^H = -13.6$$ eV.

For singly ionized carbon, we treat it as the hydrogen-like ion C$$^{5+}$$ with $$Z = 6$$. Its energy levels are $$E_n = -\frac{13.6 \times 36}{n^2}$$ eV. Setting this equal to $$-13.6$$ eV gives $$\frac{13.6 \times 36}{n^2} = 13.6$$, so $$n^2 = 36$$, hence $$n = 6$$.

The correct answer is option 2: $$n = 6$$.