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MATH / If you continue this pattern in which each line-segment extremity is replaced by…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

If you continue this pattern in which each line-segment extremity is replaced by a gradually smaller Y in the next figure, in the manner shown, how many endpoints will Figure 5 have?

[asy]
draw((0,0)--(0,-3),linewidth(.75));
draw((0,0)--(-2,2),linewidth(.75));
draw((0,0)--(2,2),linewidth(.75));
label("Figure 1",(0,-3),S);

draw((5,0)--(5,-2),linewidth(.75));
draw((4,-3)--(5,-2)--(6,-3),linewidth(.75));
draw((4,1)--(5,0)--(6,1),linewidth(.75));
draw((3,1)--(4,1)--(4,2),linewidth(.75));
draw((6,2)--(6,1)--(7,1),linewidth(.75));
label("Figure 2",(5,-3),S);

draw((10,0)--(10,-2),linewidth(.75));
draw((9.5,-2.5)--(10,-2)--(10.5,-2.5),linewidth(.75));
draw((9,-2.5)--(9.5,-2.5)--(9.5,-3),linewidth(.75));
draw((11,-2.5)--(10.5,-2.5)--(10.5,-3),linewidth(.75));

draw((9,1)--(10,0)--(11,1),linewidth(.75));
draw((8.5,1)--(9,1)--(9,1.5),linewidth(.75));
draw((11.5,1)--(11,1)--(11,1.5),linewidth(.75));
draw((8.25,.75)--(8.5,1)--(8.25,1.25),linewidth(.75));
draw((8.75,1.75)--(9,1.5)--(9.25,1.75),linewidth(.75));
draw((10.75,1.75)--(11,1.5)--(11.25,1.75),linewidth(.75));
draw((11.75,1.25)--(11.5,1)--(11.75,.75),linewidth(.75));
label("Figure 3",(10,-3),S);

[/asy]

level

Level 2

type

Algebra

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Source and history

task_id
41c68586-a1a9-5008-ac55-f7f28244095b
task_key
algebra--test--41c68586-a1a9-5008-ac55-f7f28244095b
task_revision_id
1
config / split
algebra / test

Official source

initial import