Let P(x) denote a polynomial of degree n such that P(k)= k/(k+1) for k= 0,1,2,…,n. Determine P(n+1).

Let f(x)=(x+1)P(x)-x. So f(k)=0 for k=0,1,2, ...,n. -->

f(x)=cx(x-1)(x-2)...(x-n) with f(-1)=(-1)^(n+1)*c*(n+1)!=1

So f(x)=(-1)^(n+1)*x(x-1)(x-2)...(x-n)/(n+1)!

f(n+1)=(-1)^(n+1)=(n+2)P(n+1)-(n+1) -->

P(n+1)=( (-1)^(n+1) + (n+1) )/(n+2)

Posted by Dennis on 2007-02-06 13:34:19