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The tail of a giant kangaroo is attached by a giant rubber band to a stake in the ground. A flea is sitting on top of the stake eyeing the kangaroo (hungrily). The kangaroo sees the flea leaps into the air and lands one mile from the stake (with its tail still attached to the stake by the rubber band). The flea does not give up the chase but leaps into the air and lands on the stretched rubber band one inch from the stake. The giant kangaroo, seeing this, again leaps
into the air and lands another mile from the stake (i.e., a total of two miles from the stake). The flea is undaunted and leaps into the air again, landing on the rubber band one inch further
along. Once again the giant kangaroo jumps another mile. The flea again leaps bravely into the air and lands another inch along the rubber band. If this continues indefinitely, will the flea
ever catch the kangaroo? (Assume the earth is flat and continues indefinitely in all directions.)

Okay so I tried this problem, and initially, I thought it was definitely a yes because every time the kangaroo jumps, the flea or whatever is dragged along with it. And eventually the flea will catch up the the kangaroo right? But then I tried it out in a Microsoft Excel document where
A is the #'s 1-100,
B is 5280*L1,
C is A/A*C+1 (i.e. C2 is A2/A1*C1+1 which is the ratio of the rubber band length before and after the kangaroo jumps [how much the flea is dragged along] plus the 1 inch that the flea jumps), and
D is B-C (for how far apart the flea and the kangaroo are)

The problem is, the difference between the flea and the kangaroo kept increasing and increasing so I have no idea what to do now. I know the answer is almost definitely yes... (otherwise what's the point of the problem?) but I can't get the math to work out! Is there a better way to prove it??

  • Math -

    Let the ratio of 1in/1mi be x

    After each jump, we have the fraction of the distance of the flea from the roo:


    The flea is always 1/x of the way to the roo. He'll never make it.

  • Math -

    But after each time the kangaroo jumps, the flea also jumps one inch so the ratio can't just be 1 in/1 mile

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