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What are the $\Delta E$'s of the transitions of an electron from $n=5$ to $n=1$ and from $n=5$ to $n=2$ in a Bohr hydrogen atom? The wavelength of the first electron transition is $\lambda_1=409~\mathrm{nm}$ and the other electron transition $\lambda_2=1091~\mathrm{nm}$.

How can I solve this using those two given wavelengths?

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When an electron transitions from a higher energy state to a lower energy state, the energy has to go somewhere. Usually it is emitted as light of a given wavelength which corresponds directly to the change in energy of the electron.

The energy of light at a given frequency $\nu$ is $$ E = h\nu\; , $$ where $h$ is Planck's constant. The frequency of light is directly related to the wavelength $\lambda$ and the speed of light $c$ via $$ c = \nu\lambda\; . $$

Combining these two equations we get $$ E = \frac{hc}{\lambda}\; , $$ with which you should be able to calculate the $\Delta E$ values for the transitions.

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