Let [x] denote the greatest integer not exceeding x and {x} = x -[x].
If n is a natural number, then the sum of all values of x satisfying the equation 2[x] = x + n{x} is

Given,

$$2[x]=x+nx$$

or, $$2[x]=x+n(x-[x])$$

or, $$2[x]=x+nx-n[x]$$

or, $$[x](2+n)=x(n+1)$$

or, $$x=\dfrac{\left[x\right]\cdot\left(n+2\right)}{\left(n+1\right)}$$

Now, $$\left[x\right]$$ can take integral values from 1 to n.

So sum of all values of $$x$$ = $$\dfrac{n+2}{n+1}$$ $$\cdot$$ (Sum of all possible values of $$\left[x\right]$$)

=$$\dfrac{n+2}{n+1}\left(1+2+3+4+....+n\right)$$

=$$\dfrac{n+2}{n+1}\times\ \dfrac{n\left(n+1\right)}{2}$$

=$$\dfrac{n\left(n+2\right)}{2}$$