Bouncing ball and Sum to infinity

[S_100]

"A ball is dropped vertically from a height h onto a flat surface. After the
nth bounce it returns to a height h/(3^n)
Find the total distance travelled
by the ball."

My question is Why is the sum to infinity used as opposed to Sum to n?

Total Distance = h + 2*Sum of Geometric progression (to infinity)

h + 2*h/3 / 1-1/3
h + 2h/3 *3/2 = h + h = 2h

At first I did sum to infinity purely as it would give a neater answer excluding the variable n. Then I thought this might not be accurate as energy losses will mean the ball eventually comes to a stop. But the question was posed in the maths section of a paper. So should an assumption be, the ball suffers no energy loss and thus keeps on bouncing for ever?

How can I deduce that the sum to infinity must be used from the question?

 

[MarkFL]

Realistically, we wouldn't sum to infinity because at some point a bouncing ball will come to rest. But, for the purposes of this problem, in the absence of any information regarding non-conservative forces, we are to assume the ball never comes to rest. I would say you answered the problem as intended by the author.

 

[Steven G]

My question is Why is the sum to infinity used as opposed to Sum to n?

The reason is after the nth bounce the ball is still bouncing. After all, after the nth bounce the height of the ball is h/(3^n) and this is not 0. If you stop at the nth step (besides what is n equal to? 100?, 10000?...) you did not find the total the ball has travelled.

 

[Deleted member 4993]

"A ball is dropped vertically from a height h onto a flat surface. After the
nth bounce it returns to a height h/(3^n)
Find the total distance travelled
by the ball."

My question is Why is the sum to infinity used as opposed to Sum to n?

Total Distance = h + 2*Sum of Geometric progression (to infinity)

h + 2*h/3 / 1-1/3
h + 2h/3 *3/2 = h + h = 2h

At first I did sum to infinity purely as it would give a neater answer excluding the variable n. Then I thought this might not be accurate as energy losses will mean the ball eventually comes to a stop. But the question was posed in the maths section of a paper. So should an assumption be, the ball suffers no energy loss and thus keeps on bouncing for ever?

How can I deduce that the sum to infinity must be used from the question?

Your work is correct - however you need to use grouping symbols to guide the reader to the correct solution. I would have written your as follows:

h + 2*h/3 / (1-1/3)
h + 2h/3 *3/2 = h + h = 2h

Without those (), you would get [5h/3 - 1/3]............ totally different.