How can I calculate this sum?

Question

I'm confused about how to evaluate this sum.

It looks complicated to me but I should know how to calculate it......thank you.

Answer 1

$\sum_\limits{i=1}^n \left( -2 \times\frac {1}{n} + 3\times \frac {1}{n}\right)$

First we can add the fractions together.

$\sum_\limits{i=1}^{n-1} \frac {1}{n}$

Since $\frac {1}{n}$ does not depend on $i$ we can factor that out. Not entirely necessary, but it should make it a little more clear.

$\frac 1n \sum_\limits{i=1}^{n-1} 1$

$\sum_\limits{i=1}^{n-1} 1 = n-1$

$\frac 1n \sum_\limits{i=1}^{n-1} 1 = \frac {n-1}{n}$