445. A table of values of a known function $f(n)$ for natural values of $n$ has been compiled; it turned out that the second difference is constantly equal to zero. By what formula can $f(n)$ be represented for any natural $n$? How does the situation stand for fractional values of $n$?
445. From the fact that the second differences are zero, it follows that the first differences are constant; let us denote them by $a$, so $f(2)-f(1)=a, f(3)-f(2)=a, \ldots f(n)-f(n-1)=a$. Adding these $n-1$ equations side by side, we have $f(n)-f(1)=(n-1) a$, or, denoting $f(1)$ by the letter $b, f(n)=(n-1) a+b=n a+b-a$. Thus, for natural numbers $n$, the function $f(n)$ is a linear function of $\boldsymbol{n}$; but nothing allows us to judge what values $f(n)$ takes for non-integer $n$.