Coordinated Trouble #1

Geometry Level 4

A B C D . . . . a r e n p o i n t s i n a p l a n e w h o s e c o o r d i n a t e s a r e ( 1 , 2 ) ( 8 , 16 ) , ( 27 , 54 ) ( 64 , 128 ) ( 125 , 250 ) . . . ( 729 , 1458 ) . A B i s b i s e c t e d i n t h e p o i n t G 1 ; G 1 C i s d i v i d e d a t G 2 i n t h e r a t i o 1 : 2 ; G 2 D i s d i v i d e d a t G 3 i n t h e r a t i o 1 : 3 ; G 3 E a t G 4 i n t h e r a t i o 1 : 4 a n d s o o n u n t i l a l l t h e p o i n t s a r e e x h a u s t e d . I f t h e c o o r d i n a t e o f t h e f i n a l p o i n t i s ( X Y ) . F i n d Y X . 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 } .


The answer is 2.

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3 solutions

Patrick Corn
Sep 8, 2014

Call me silly, but all the points in the construction are on the line y = 2 x y = 2x . So the ratio you want is 2 2 .

Nice Patrick Corn .Yes there is that way,too. But there's also a rigorous one.

Soummo Mukherjee - 6 years, 9 months ago

All the points lie on the line y=2x

Shubham Agrawal
Sep 25, 2014

Quite easy as that paoint will also be on y = 2x...btw X = 225.

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