If $f(x)$ denotes a polynomial of degree $n$ such that $f(k) =\dfrac1k$ for $k = 1,2,3,\ldots,(n+1)$, determine $f(n + 2)$.

Question

Let $f(x)$ be a polynomial of degree $n$ such that $$f(k) =\dfrac1k$$ for $k = 1,2,3,\ldots,(n+1)$. Determine $f(n + 2)$.

This question is quite similar to this one. How do I solve such type of problems in general?

Answer 1

Hint: Note that we have $kf(k) = 1$ for $k = 1,\ldots,n+1$, and clearly $0f(0) = 0$. So now we know $n+2$ values of the $(n+1)$-degree polynomial $xf(x)$.

As for the general bit, it's difficult to say. You should try to make everything into polynomials as nicely as possible, because polynomials are a lot easier to work with than rational functions. Case in point: I can't tell much about $f(x)$ just from knowing the solutions of $f(x) = \frac1x$, but I can tell a lot about $xf(x)$ from knowing the solutions to $xf(x) = 1$.

It won't always work, though, and the most important thing, as with problem solving in general, is to just try things and not give up when your first attempt fails. Remember that when you read a solution that someone else has written to a problem like this, you only see the attempts that succeeded. Behind each successful solution are either a lot of failed attempts at solving the problem, or the experience needed to write a solution more or less on the first attempt, built up through a lot of trial and error on other, similar problems.

Answer 2

Let $g(x):=x\,f(x)$ like Arthur suggests. Then, $g$ is of degree $n+1$. Hence, from this link, we have $$\sum_{r=0}^{n+2}\,(-1)^r\,\binom{n+2}{r}\,g(x+r)\equiv 0\,.$$ In particular, $$\sum_{r=0}^{n+2}\,(-1)^r\,\binom{n+2}{r}\,g(r)=0\,.$$ As $g(0)=0$ and $g(1)=g(2)=\ldots=g(n+1)=1$, we conclude that $$\begin{align}g(n+2)&=(-1)^{n+1}\,\sum_{r=1}^{n+1}\,(-1)^{r}\,\binom{n+2}{r} \\&=(-1)^{n+1}\,\left((1-1)^{n+2}-1-(-1)^{n+2}\right)=(-1)^{n}+1\,.\end{align}$$ Consequently, $$f(n+2)=\frac{1+(-1)^n}{n+2}\,.$$