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Solve Sequences Used in Given Situation

There is a rope that stretches from the top of Maidwell building to a tree on the racecourse, and the length of this rope is 1km. A worm begins to travel along the rope at the rate of 1cm each second in an attempt to get to the other end. then a strange thing happens...

some malevolent deity intervenes to make life even harder for this poor little creature. every second (starting one second after the worm begins its journey) the rope is stretched so as to become 1 km longer. (So 0,1,2,3 seconds after the start the rope has length 1km, 2km,3km,4km, and so on.) In between these instantaneous stretchings the worm continues his 1cm per second crawll. The stretchings are uniform along the length of the rope, so there is a stretching of the distance of the worm from the start as well as a stretching of his distance from the far end. But each stretching maintains the proportion of the total that the worm has travelled.

so what happens? how far along the rope can the worm get? how long does it take him?

[Hint; because each stretch preserves the proportion of the total distance that the worm has travelled, it's best to do all the calculations az fractions of the total, rather than deal with actual distance.]

i know that there is a sequence for the length of rope 1,2,3,4,....
also a sequence for the time taken (second)0,1,2,3.......
but i don't know how to go about solving such sequence as applying to the stated situation above.

Solution Preview

Hi there. This is one of those tricky things here, isn't it?

The clue is given to look at *not* the total but the fraction of the distance travelled.

First second: worm travels 1 cm out of 1 km = 1 part in 100 x 1000 (100 cm per meter, 1000 m per km) = 1 part in ...