Coordinated Trouble #1

Geometry Level 4

\( A\quad B\quad C\quad D....\quad are\quad n\quad points\quad in\quad a\quad plane\\ whose\quad coordinates\quad are\quad \left( 1,2 \right) \quad \left( 8,16 \right) ,\quad \left( 27,54 \right) \quad \left( 64,128 \right) \quad \left( 125,250 \right) ...\left( 729,1458 \right) .\quad \\ AB\quad is\quad bisected\quad in\quad the\quad point\quad { G }_{ 1 };\quad { G }_{ 1 }C\quad is\\ divided\quad at\quad { G }_{ 2 }\quad in\quad the\quad ratio\quad 1:2;\quad { G }_{ 2 }D\quad is\quad \\ divided\quad at\quad { G }_{ 3 }\quad in\quad the\quad ratio\quad 1:3;\quad { G }_{ 3 }E\quad \\ at\quad { G }_{ 4 }\quad in\quad the\quad ratio\quad 1:4\quad and\quad so\quad on\quad until\quad \\ all\quad the\quad points\quad are\quad exhausted.\quad If\quad the\quad \\ coordinate\quad of\quad the\quad final\quad point\quad is\quad (X\quad Y).\\ Find\quad \frac { Y }{ X } . \)

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