Roots

Newton's Sum: $\large S_{100} - 6S_{99} - 2 S_{98} = 0 \Rightarrow \frac {S_{100}-2S_{98}}{S_{99}} = 6$

{ x }^{ 2 }-6x-2\quad =\quad 0\ \Longrightarrow { \alpha }^{ 2 }-2\quad =\quad 6\alpha \quad and\quad { \beta }^{ 2 }-2\quad =\quad 6\beta \ \Longrightarrow { \alpha }^{ 100 }+{ \beta }^{ 100 }\quad -\quad 2 { \alpha }^{ 98 }-2 { \beta }^{ 98 }\quad =\quad { \alpha }^{ 98 }({ \alpha }^{ 2 }-2)+{ \beta }^{ 98 }(\beta ^{ 2 }-2)\quad =\quad 6({ \alpha }^{ 99 }+{ \beta }^{ 99 })\ therefore\quad \xleftarrow [ { \alpha }^{ 100 }+{ \beta }^{ 100 }\quad -\quad 2 { \alpha }^{ 98 }-2 { \beta }^{ 98 } ]{ { \alpha }^{ 99 }+{ \beta }^{ 99 } } =\quad \xleftarrow [ 6({ \alpha }^{ 99 }+{ \beta }^{ 99 }) ]{ { \alpha }^{ 99 }+{ \beta }^{ 99 } } =\quad 6