Polynomial

#Problem 14

Let f be a polynomial of degree 98, such that f(k)=1/k. [k=1,2,3,4...99]. Find the value of f(100).[USAMath Talent Search 2009]

Solution scheme and approach
Consider a function g(k) such that
g(k)=kf(k)-1
As f(k) is a polynomial of deg 98 g(k) should be a polynomial of deg 99.
=>g(k)=C*(k-1)*(k-2)*(k-3)*...*(k-99)
=>kf(k)-1=C*(k-1)*(k-2)*(k-3)*...*(k-99)............(1)
by putting k=0
=>C=1/99!
=>kf(k)-1=[(k-1)*(k-2)*(k-3)*...*(k-99)]/99!
If k=100
=>100f(100)-1=1
=>f(100)=1/50