An isotope of the original nucleus can be formed in a radioactive decay, with the emission of following particles.

Two different radioactive elements with half lives '$$\mathrm{T}_1$$' and '$$\mathrm{T}_2$$' have undecayed atoms '$$\mathrm{N}_1$$' and '$$\mathrm{N}_2$$' respectively present at a given instant. The ratio of their activities at that instant is

In Balmer series, wavelength of the $$2^{\text {nd }}$$ line is '$$\lambda_1$$' and for Paschen series, wavelength of the $$1^{\text {st }}$$ line is '$$\lambda_2$$', then the ratio '$$\lambda_1$$' to '$$\lambda_2$$' is

In Lyman series, series limit of wavelength is $$\lambda_1$$. The wavelength of first line of Lyman series is $$\lambda_2$$ and in Balmer series, the series limit of wavelength is $$\lambda_3$$. Then the relation between $$\lambda_1$$, $$\lambda_2$$ and $$\lambda_3$$ is