### Instruction: Let the term $S_n$ be the sum of the first $n$ powers of $2$. For instance, $S_3 = 2^0 + 2^1 + 2^2 = 7$.  Find the largest possible value of the greatest common divisor of two consecutive terms, $S_n$ and $S_{n+1}$, for any $n$.### Response: I notice that each term $S_n$ is one less than the next power of $2$, since $S_n = 2^{n+1} - 1$. So I can rewrite the terms as $S_n = 2^{n+1} - 1$ and $S_{n+1} = 2^{n+2} - 1$.
To find the greatest common divisor of two numbers, I can use the Euclidean algorithm, which repeatedly applies the division algorithm.
So I start by dividing $S_{n+1}$ by $S_n$, and get a quotient of $2$ and a remainder of $1$. That is, $S_{n+1} = 2S_n + 1$.
This means that the greatest common divisor of $S_n$ and $S_{n+1}$ is the same as the greatest common divisor of $S_n$ and $1$.
But the greatest common divisor of any number and $1$ is just $1$.
So the greatest common divisor of $S_n$ and $S_{n+1}$ is $1$, regardless of $n$.
Therefore, the largest possible value of the greatest common divisor of two consecutive terms is $1$.