Convolution of Two Impulse Signals I have encountered convolution of two different impulse signals.

x[n] = (1/2)^n . u[n-2] * u[n]
x[n] = u[n] * [n]

u[n] = discrete impulse signal
. = product operation
* = convolution operation


For the first one, I found this solution:

x[n] = 1/4 if n = 2
x[n] = 0 if n != 2


For the second one, I found impulse signal itself 

Edit: Are my answers are true ? My professor told me that the answer for the first one is wrong, but he did not say the correct answer.
Your answer is right, assuming you posted the question right. 

But you better use the standard notation as Dilip Sarwate already indicated; $u[n]$ is the unit-step and $\delta[n]$ is the unit impulse. Then

$$ 0.5^n \delta[n-2] \star \delta[n] = 0.5^2 \delta[n-2] = \begin{cases} { 0.25 ~~~, ~~~n= 2 \\ 0.00 ~~~,~~~n \neq 2 } \end{cases}
$$

you can get the answer for the second case, exactly in the same way.