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Find f(2) if f(1)=8 and f(4)= 27

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The value of f(5) is 13.

To find the value of f(5) given that f(1) = 1, f(2) = 2, f(3) = 3, and f(4) = 16, we need to determine the polynomial function f(x).

Since the function f(x) is a polynomial, we can use the method of finite differences to determine the degree of the polynomial. The first differences are:

1 2 3 16

1 1 13

0 12

The second differences are constant, which means that the polynomial is a quadratic. We can use the values of f(1), f(2), and f(3) to write the quadratic in the form:

f(x) = ax^2 + bx + c

where a, b, and c are constants. Substituting the given values, we get:

1 = a + b + c

2 = 4a + 2b + c

3 = 9a + 3b + c

Solving this system of equations, we get:

a = 1

b = -3

c = 3

Therefore, the polynomial function is:

f(x) = x^2 - 3x + 3

Substituting x = 5, we get:

f(5) = 5^2 - 3(5) + 3 = 25 - 15 + 3 = 13

Therefore, the value of f(5) is 13.

Complete question:

If f(x) be a polynomial such that f(1)=1,f(2)=2,f(3)=3 and f(4)=16.Find the value of f(5).

User Stephen Kennedy
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