Can somebody teach me how to solve this question? Please…
Sorry... since i don't know how to write the symbols here... so i used attachment...
If it's $\displaystyle \sum_{r = 1}^n \frac{r^2 + r^{-1}}{r^2 + r} = \frac{n^2}{n+1}$, then it isn't even true.
$\displaystyle n =1:$ LHS $\displaystyle = \frac{1^2 +1^{-1}}{1^2 + 1} = \frac{2}{2} = 1$. RHS $\displaystyle = \frac{1^2}{1+1} = \frac{1}{2}$.
$\displaystyle n =2$: LHS $\displaystyle = (1) + \frac{2^2 +2^{-1}}{2^2 + 2} = 1 + \frac{4.5}{6} = \frac{7}{4}$. RHS $\displaystyle = \frac{2^2}{2+1} = \frac{4}{3}$.
And yet you had to add the phrase "only the final step will convert r to n," the meaning of which I don't understand. Even worse, the left-hand side of the equation in the quote in post #2 does not depend on n (n is a bound variable used as the summation index) while the right hand side does. And if n and r are reversed, then, as post #3 shows, the equation is incorrect.