I am not sure how the greatest common divisor of one number is defined. It can be $1$, or first greater number than $1$, or that number, or first smaller number from that given number.
If $n=1\Rightarrow 5n-2=3\Rightarrow\sum_\limits{k=0}^{5n-2}=15$.
The first number greater than $1$ that divides $15$ is $3$, and the first smaller than $15$ is $5$.
If $n=2\Rightarrow 5n-2=8\Rightarrow\sum_\limits{k=0}^{5n-2}=511$.
The first number greater than $1$ that divides $511$ is $7$, and the first smaller than $511$ is $73$.
Is it possible to find a property for greatest common divisor of $\sum_\limits{k=0}^{5n-2}2^k$ in both cases (first greater than $1$, and the second, smaller than that number)?