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# AMC 10B Question (i just took it)

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Henry decides one morning to do a workout and walks $$\frac{3}{4}$$ of the way from his home to his gym. The gym $$2$$ kilometers away from Henry's home. At that point, he changes his mind and walks $$\frac{3}{4}$$ of the distance from there back toward the gym. If Henry keeps changing his mind when he has walked $$\frac{3}{4}$$of the distance toward either the gym or home from the point where he last changed his mind, he will get very close to walking back and forth between a point $$A$$ kilometers from home and a point $$B$$ kilometers from home. What is $$|A-B|$$ ?

Feb 14, 2019

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He will be 2(3/4 - 9/16 + 27/64 - 81/256 + ... ) meters from his home finally.

This is a G.S. with first term 3/2 and common ratio -3/4. Sum of the G.S. (i.e. the final distance from home) is $$2\left(\dfrac{\dfrac{3}{4}}{1-\left(\dfrac{-3}{4}\right)} \right) = \dfrac{6}{7}$$.

So A = 6/7, B = 2 - 6/7 = 8/7.

|A - B| = |(6-8)/7| = $$\dfrac{2}{7}$$.

Feb 14, 2019
edited by MaxWong  Feb 14, 2019