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 $n=1$: $LHS=F_1^2=1$, $RHS=F_1F_2=1$, thus it is true for $n=1$.
Step 3. Assume the statement is true when $n=k$: we have $F_1^2+F_2^2+F_3^2+...+F_k^2=F_kF_{k+1}$.
Step 4. Prove it is true for $n=k+1$: $LHS=F_1^2+F_2^2+F_3^2+...+F_k^2+F^2_{k+1}=F_k F_{k+1}+F^2_{k+1}=F_{k+1}(F_k+F_{k+1})$, and $RHS=F_{k+1} F_{k+2}=F_{k+1}(F_k+F_{k+1})=LHS$
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 $n$.
