I have a finite summation series of identical fractions e.g $ \frac{1}{3} + \frac{1}{3} + \frac{1}{3} + \frac{1}{3} + \frac{1}{3} = \frac{5}{3}$. Now lets say I add one to the denominator for the first two values of the series. I now have $ \frac{1}{4} + \frac{1}{4} + \frac{1}{3} + \frac{1}{3} + \frac{1}{3} = \frac{3}{2}$. Now I do it again but this time to the first three values of the series. $ \frac{1}{5} + \frac{1}{5} + \frac{1}{4} + \frac{1}{3} + \frac{1}{3} = \frac{79}{60}$. To complicate things again I add one to the denominator of the last three values of the series too. $ \frac{1}{5} + \frac{1}{5} + \frac{1}{5} + \frac{1}{4} + \frac{1}{4} = \frac{11}{10}$
Is there an easy direct method to figure out the final answer $\frac{11}{10}$ given that we know that we did $[2,3]$ additions to the left of the series, and $[3]$ additions to the right of the series?