Fibonacci squares

Algebra Level 2

1,1,2,3,5,8,13,21,34,55,89,\large 1,1,2,3,5,8,13,21,34,55,89, \cdots

The above is the Fibonacci sequence, where FnF_n is the nnth Fibonacci number with F1=F2=1F_1 = F_2 = 1.

Without using a calculator, find which answer is equivalent to (F10)2(F9)2{ ({ F }_{ 10 }) }^{ 2 }-{ ({ F }_{ 9 }) }^{ 2 }.

Hint: Try to find a simpler formula for (Fn+1)2(Fn)2{ ({ F }_{ n+1 }) }^{ 2 }-{ ({ F }_{ n }) }^{ 2 }.

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