Continuous or discrete distribution function

[eboolean]

k is a positive integer. Find the distribution function of the random variable X.

I tried some solution I cant figure it out. I shared with you my solutions. Where am I making a mistake? should this function be Continuous or discrete?

 

[pka]

View attachment 20000
k is a positive integer. Find the distribution function of the random variable X. I tried some solution I cant figure it out. I shared with you my solutions. Where am I making a mistake? should this function be Continuous or discrete?

It is a shame that you have not been instructed about the differences in continuous and discrete.
Are these events countable or not? I know the are countable, hence discrete.
\(\forall k\left[P(X=k)=\left(\dfrac{1}{2}\right)^k>0\right]~\&~\displaystyle\sum\limits_{k = 1}^\infty {{{\left( {\frac{1}{2}} \right)}^k}}=1 \) SEE HERE
So you are really and inexcusably confused on these basic basic concepts.
What is the question, please please do tell us what is the question.