It is known that for any natural number n, f (n + 1) + F (n-1) = 2F (n), where f (0) ≠ 0, f (1) = 1

Isn't that the definition of arithmetic sequence? What's the question?
f(n+1)-f(n)=f(n)-f(n-1)=f(1)-f(0)=d
f(n)=f(1)+(n-1)d=1+(n-1)d

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