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Question Number 62322 by aliesam last updated on 19/Jun/19

Commented by arcana last updated on 19/Jun/19

$${\beta }_1=\alpha +{\alpha }^6$$$${\beta }_2={(\alpha +{\alpha }^6)}^2-2={\alpha }^2+2{\alpha }^7+{\alpha }^{12}-2$$$${\beta }_2={\alpha }^2+{\alpha }^5$$$${\beta }_3={({\alpha }^2+{\alpha }^5)}^2-2={\alpha }^4+2{\alpha }^7+{\alpha }^{10}-2$$$${\beta }_3={\alpha }^4+{\alpha }^3$$$${\beta }_4={({\alpha }^4+{\alpha }^3)}^2-2={\alpha }^8+2{\alpha }^7+{\alpha }^6-2$$$${\beta }_4=\alpha +{\alpha }^6={\beta }_1$$$$\Rightarrow \underset{1}{\sum ^{99}}{\beta }_k=33\underset{1}{\sum ^3}{\beta }_k=33[{\beta }_1+{\beta }_2+{\beta }_3]$$$$=33[\alpha +{\alpha }^6+{\alpha }^2+{\alpha }^5+{\alpha }^4+{\alpha }^3]=-33$$$$$$$$$$

Commented by alphaprime last updated on 19/Jun/19

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