**#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

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