How do I calculate the change in energy of an electron transition? 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?
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.