I am confused at a step in the proof of Cauchy Criterion otherwise known as Cauchy Condensation
Solution 1:
In the sum$$\sum_{n=2^K+1}^{2^{K+1}}a_{2^{K+1}},\tag1$$you are summing $2^K$ numbers, each of which is equal to $a_{2^{K+1}}$. Therefore $(1)$ is equal to $2^Ka_{2^{K+1}}$.