$\mathrm{\omega }+2{\mathrm{\omega }}^2+3{\mathrm{\omega }}^3+......90{\mathrm{\omega }}^{90}=$

$\begin{array}{rcl} & & \mathrm{w}+2{\mathrm{w}}^2+3{\mathrm{w}}^3+....+90{\mathrm{w}}^{30}\\ \mathrm{w}& =& {\mathrm{w}}^4={\mathrm{w}}^7=....={\mathrm{w}}^{88}\\ {\mathrm{w}}^2& =& {\mathrm{w}}^5={\mathrm{w}}^8=...={\mathrm{w}}^{89}\\ {\mathrm{w}}^3& =& {\mathrm{w}}^6= ={\mathrm{w}}^{90}\\ & =& \mathrm{w}(1+4+7+...+88)+{\mathrm{w}}^2(2+5+...+89)+{\mathrm{w}}^3(3+6+...+90)\\ & =& \mathrm{w}·\frac{30}{2}(1+88)+{\mathrm{w}}^2·\frac{30}{2}(2+89)+{\mathrm{w}}^3(3+90)\times \frac{30}{2}\\ & =& 15(89\mathrm{w}+91{\mathrm{w}}^2+93)\\ & =& 15\left(89\right(1+\mathrm{w}+{\mathrm{w}}^2)+2{\mathrm{w}}^2+4)\\ & =& 15\left(2\right(1+{\mathrm{w}}^2)+2)\\ & =& 15(-2\mathrm{w}+2)\\ & =& 30(1-\mathrm{w})\end{array}$