If $\omega$ is a complex cube root of unity, then the value of $\sin\left[\pi(\omega^{10} + \omega^{23}) - \dfrac{\pi}{4}\right] =$

If $\omega$ is a complex cube root of unity, then the value of the expression $2\left(1+\dfrac{1}{\omega}\right)\left(1+\dfrac{1}{\omega^2}\right) + 3\left(2+\dfrac{1}{\omega}\right)\left(2+\dfrac{1}{\omega^2}\right) + \ldots + (n+1)\left(n+\dfrac{1}{\omega}\right)\left(n+\dfrac{1}{\omega^2}\right)$ is...

The area of the triangle whose vertices are $i, \omega$ and $\omega^2$ is (Where $\omega$ is a complex cube root of unity other than $1, i$ is an imaginary number)__________ sq.units