If the interpolation of $f(x)$ on the set of distinct points $x_0, x_1, \cdots x_n$ is given by $$\sum_{k=0}^{n} l_k(x)f(x_k).$$ Find an expression for $$\sum_{k=0}^{n} l_k(0)x_k^{n+1}.$$
I don't know how to prove the problem. It seems to me that $\sum_{k=0}^{n} l_k(x)f(x_k)$ is like the Lagrange's interpolation formula and we have to find a suitable form of $f(x)$ such the expression $\sum_{k=0}^{n} l_k(0)x_k^{n+1}$ will come out from $\sum_{k=0}^{n} l_k(x)f(x_k)$. But don't know how to get the suitable form.
The required expression is the value at $0$ of the Lagrange interpolation polynomial $g(x)$ for $f(x) = x^{n+1}$. The function $g(x)$ is the unique polynomial of degree $\leq n$ that agrees with $f(x)$ at $x_0, x_1, \dots, x_n$.
Obviously, $g(x) = x^{n+1} - (x-x_0)(x-x_1) \dots (x - x_n)$ satisfies the required conditions, so it must be the correct choice of $g(x)$.
The answer is $g(0) = (-1)^n x_0 x_1 \dots x_n$.