Answer
See explanations.
Work Step by Step
Step 1. Recall the property of Fibonacci sequence, we have: $F_{n}=F_{n-1}+F_{n-2}$ and $F_1=F_2=1$
Step 2. Prove the statement is true when $n=1$: $LHS=F_1=1$, $RHS=F_3-1=F_2+F_1-1=1$, thus it is true for $n=1$.
Step 3. Assume the statement is true when $n=k$: we have $F_1+F_2+F_3+...+F_k=F_{k+2}-1$
Step 4. Prove it is true for $n=k+1$: $LHS=F_1+F_2+F_3+...+F_k+F_{k+1}=F_{k+2}-1+F_{k+1}=F_{k+2}+F_{k+1}-1$, and $RHS=F_{k+3}-1=F_{k+2}+F_{k+1}-1$ which gives $LHS=RHS$
Thus, the statement is also true for $n=k+1$
Step 5. Conclusion: with mathematical induction, we have proved that the statement is true for all natural numbers $n$.
