It is given that $\displaystyle U_1 =e^2 , U_{r+1} - U_r = (e-10)e^{r+1}$
By considering $\displaystyle \sum^{n}_{r=1}(U_{r+1} - U_r , show that, U_n =e^{n+1} $
I dont know do the summation
You are told that $\displaystyle U_{r+1}- U_r= (e- 10)e^{r+1}= e(e-10)e^r$
so $\displaystyle \sum_{r=1}^n(U_{r+1}- U_r)= e(e-10)\sum_{r=1}^n e^r$.
$\displaystyle \sum_{r=1}^n e^r= \sum_{r=0}^n e^r- 1$
That's a geometric series. Do you know the formula for the sum of a geometric series?