But I'm Not Listing All Of Them!

Relevant wiki: Fibonacci Sequence

Let $\mathbf{S_n}=\displaystyle\sum_{r=1}^nf_n$ .

$\begin{aligned}\mathbf{S_n}=&1+1+2+\cdots +\color{#D61F06}{f_n}\\\mathbf{S_n}=&~~~~~~~1+1+\cdots+\color{#D61F06}{f_{n-1}}+\color{#3D99F6}{f_n}\end{aligned}$

Adding and using $\color{#D61F06}{f_{n}+f_{n-1}}=\color{#3D99F6}{f_{n+1}}$ and then using $\color{#3D99F6}{f_{n+1}+f_{n}}=f_{n+2}$

$2\mathbf{S_n}=\underbrace{1+2+3+\cdots+f_n}_{\large \mathbf{S_n}~-1}+ f_{n+2}$

$\large \implies\mathbf{S_n}=f_{n+2}-1$

Put $n=20$ to get: $\Large \mathbf{S_{20}}=\Large\boxed{17710}$

Let's prove $\sum_{k=1}^{n}f_k=f_{n+2}-1$ by induction: $\sum_{k=1}^{n+1}f_k=f_{n+1}+\sum_{k=1}^{n}f_k=f_{n+1}+f_{n+2}-1$ $=f_{n+3}-1$ . Thus the answer is $f_{22}-1=\boxed{17710}$