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Let (F_n)_{n\geq 1} be the Fibonacci sequence F_1 = F_2 = 1, F_{n+2} = F_{n+1} + F_n (n \geq 1), and P(x) the polynomial of degree 990 satisfying
P(k) = F_k, \qquad \text{ for } k = 992, . . . , 1982.
Prove that P(1983) = F_{1983} - 1.

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