Problem - 3657
Let the domain of function $f(n)$ be $\mathbb{N}$, $f(1)=1$, and for any integer $n \ge 2$, $$f(n)=f(n-1) + 2^{n-1}$$
Determine $f(n)$.
Setting $n=2, 3, \cdots$, respectively:
\begin{align*}
f(2) &= f(1) + 2^1\\
f(3) &= f(2) + 2^2\\
\cdots\\
f(n) &= f(n-1) + 2^{n-1}
\end{align*}
Adding them together gives $$f(n) = f(1)+2^1 + 2^2+\cdots + 2^{n-1}= 1 + 21 + 2^2+\cdots + 2^{n-1}=\boxed{2^n-1}$$