Determining type of filter given its equation
The $\mathcal Z$-transform of the given difference equation is :

$$H(z) = 1 + z^{-2}$$

Put $z= \exp(j\omega)$ and remember that in discrete time systems the low frequencies are at $2n\pi$ $(n=0,1,2, \ldots)$ and the high frequencies occur at $(2n+1)\pi$; this is just a consequence of the periodic behavior of the discrete time complex exponentials.

So, $H(z) = 1 + \exp(-2j\omega)$ at $z = \exp(j\omega)$.

When $\omega=0; H(z) = 2$ and $w = \pi$ gives $H(z)= 2$.

Thus, both at high and low frequencies the the system function provides same 
gain and hence the filter with the given $H(z)$ is a BAND REJECT/ NOTCH FILTER with $H(z) = 0$ at $\omega = \pi/2$.