Stochastic process markov chain probability feature

Question

I could not understant the reason of following equality? How did they write as difference of two random variables?

Answer 1

The random variable $T_n$ is the time of $n$th success. The time to the $(n+1)$th success is equal to the time to $n$th success, plus a random number $K$.

Let $j-i=k$. Then $$T_{n+1}=T_n+k$$ This can be written as $$T_{n+1}-T_n=k=j-i$$

It follows that $K\sim \tt Geometric(p)$ whose pdf is $$f_K(k)=p(1-p)^{k-1},\,k\in\{1,2,3,\cdots\}$$