Suppose f(x) is a degree 8 polynomial such that f(2^i)=1/2^i for all integers 0≤i≤8. If f(0)=a/b, where a and b are coprime positive integers, what is the value of a+b?

To find the value of f(0) = a/b, we can use Lagrange interpolation.

Lagrange interpolation allows us to find a polynomial that passes through a set of points. In this case, we have the points (2^i, 1/2^i) for i = 0 to 8.

First, let's write down the Lagrange interpolation formula for a general polynomial of degree n:
P(x) = Σ[f(xᵢ) * L(x)] for i = 0 to n

Where L(x) is the Lagrange basis polynomial given by:
L(x) = Π[(x - xⱼ) / (xᵢ - xⱼ)] for j ≠ i, j = 0 to n

Now let's substitute the values we have into the formula:

P(x) = [(1/2^0) * L(x)] + [(1/2^1) * L(x)] + [(1/2^2) * L(x)] + ... + [(1/2^8) * L(x)]

Since we know that f(x) is a degree 8 polynomial, we can substitute P(x) = f(x) into the equation:

f(x) = [(1/2^0) * L(x)] + [(1/2^1) * L(x)] + [(1/2^2) * L(x)] + ... + [(1/2^8) * L(x)]

Now, let's substitute x = 0 into the equation to find f(0):

f(0) = [(1/2^0) * L(0)] + [(1/2^1) * L(0)] + [(1/2^2) * L(0)] + ... + [(1/2^8) * L(0)]

Since L(0) will involve divisions by zero, we need to modify our equation slightly. Let's multiply both sides by a factor of x to remove the division by zero:

f(0)x = [(1/2^0) * x * L(x)] + [(1/2^1) * x * L(x)] + [(1/2^2) * x * L(x)] + ... + [(1/2^8) * x * L(x)]

Now, substitute x = 0:

f(0) * 0 = [(1/2^0) * 0 * L(0)] + [(1/2^1) * 0 * L(0)] + [(1/2^2) * 0 * L(0)] + ... + [(1/2^8) * 0 * L(0)]

Simplifying the equation, we get:

0 = 0 + 0 + 0 + ... + 0

Therefore, f(0) = 0.

Since a and b are coprime positive integers, a + b = 0 + 1 = 1.

So, the value of a+b is 1.