The key to this is to realise that once the flea is on the rubber band, as the band stretches it carries the flea with it.
After one jump, the kangaroo has moved 1 mile and the flea is 1 inch along the band. When the kangaroo jumps for the second time, the rubber band stretches from 1 mile to 2 miles, so the flea will then be 2 inches (not 1 inch) along the band. The flea then jumps another inch. So by the time they have both jumped twice, the flea is 3 inches along the band.
When the kangaroo makes its third jump the band stretches from 2 miles to 3 miles long, carry the flea from 3 inches along to 4½ inches along the band. It then jumps another inch, bringing its total distance to 5½ inches.
Work out what happens after the next few jumps, and see if you can guess a formula for how far the flea has moved after n jumps. Then prove by induction that your guess is correct.
The kangaroo moves a fixed distance of 1 mile for each jump. But the distance moved by the flea is increasing with each jump. After a miilion or so jumps, will the flea finally reach the kangaroo?