Kevin kangaroo begins hopping on a number line at 0. he wants to get to 1, but he can hop only 1/3 of the distance. each hop tires him out so that he continues to hop 1/3 of the remaining distance. how far has he hopped after five hops? express your answer as a common fraction.

Explanation:

This is called a series, to solve it you need to give the first hop which is going to move you 1/3 of the way, the you hop another time, this will move you 1/3 of the 2/3 missing, this means you have moved now:

and you are missing 4/9 of the way.

Next hope will move you 1/3 of the 4/9 missing, which is , adding this to the path you have already moved is:

and you are missing 8/27 of the way.

The fourth hop is the same, one third of the missing path:  , and adding this to the traveled path:

and you are missing 16/81 of the way.

The last and fifth hop is again one third of the missing path: , and adding this to the already moved way:

And you end here.

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