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Let f(n) be a polynomial of degree 3 such that f(n) = 2^n for n = 0, 1, 2, 3. Find f(4).

 Feb 17, 2021
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We can write f(n) = ax^3 + bx^2 + cx + d. We can write a system of equations:

 

d = 1

a + b + c + d = 2

8a + 4b + 2c + d = 4

27a + 9b + 3c + d = 8

 

(a, b, c, d) = (1/6, 0, 5/6, 1)

 

So, \(f(4) = \frac{1}{6}(4)^3 + \frac{5}{6}(4) + 1 = \boxed{15}\)

 Feb 19, 2021

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