Finding a formula for the sums $2^0 + 2^1 + 2^2 + 2^3 + \cdots + 2^n$ I am told to consider the sum $2^0 + 2^1 + 2^2 + 2^3 + \cdots + 2^n$ for $n ∈ ℕ ∪ \{0\}$. Next, I am to find the value of the sum for $n= 0, 1, 2, 3, 4$. After I find those values, I am supposed to come up with a conjecture that gives a simple closed formula. When I did the sums, I realize that the difference between sums doubles every time. So,
$n=0$ is $1$,
$n=1$ is $3$,
$n=2$ is $7$,
$n=3$ is $15$,
$n=4$ is $31$.
My problem is that I am having a hard time trying to come up with a conjecture. How would I be able to come up with one? Is it just Guess and Check?

As the difference doubles every (they are 2, 4, 8, 16) we have $x_{n+1}=x_n+2^{n+1}$. So using this we can see
$$x_n=\sum_{n=0}^n2^{n}$$
This is a finite geometric series, so it is given by
\begin{align*}x_n=\sum_{n=0}^n2^{n}&=\frac{1-2^{n+1}}{1-2} \\&=2^{n+1}-1\end{align*}
