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).