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Question
F is a polynomial function of degree 6 such that f(n)=1/n For n= 1,2,3,4,5,6,7. Find f(8)

Answer
(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)(x-7) =


x^7 - 28x^6 + 322x^5 - 1960x^4 + 6769x^3 - 13132x^2 + 13068x - 5040



so


n^7 - 28n^6 + 322n^5 - 1960n^4 + 6769n^3 - 13132n^2 + 13068n = 5040


for n = 1,2,3,4,5,6,7


Divide both sides by 5040


(1/5040)n^7 - (28/5040)n^6 + (322/5040)n^5 - (1960/5040)n^4 + (6769/5040)n^3 - (13132/5040)n^2 + (13068/5040)n = 1

Divide both sides by n

(1/5040)n^6 - (28/5040)n^5 + (322/5040)n^4 - (1960/5040)n^3 + (6769/5040)n^2 - (13132/5040)n + 13068/5040 = 1/n




so f(x) = (1/5040)x^6 - (28/5040)x^5 + (322/5040)x^4 - (1960/5040)x^3 + (6769/5040)x^2 - (13132/5040)x + 13068/5040


5040f(x) = x^6 - 28x^5 + 322x^4 - 1960x^3 + 6769x^2 - 13132x + 13068


Evaluate the right side at x = 8

5040f(8) = 1260


f(8) = 1260/5040 = 1/4


the answer is 1/4

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