
rubber ball dropped
A rubber ball dropped from a height of 10 meters, rebounded 3/4 of it's height from which it fell. If it continued to bounce and rebound so that each new height was 3/4 the previous height, how far did the ball travel before hitting the ground 20 times?

The nth partial sum of a geometric series is
Where
The infinite series can be gotten from

y can't we take the sum of distance travelled by the ball after hitting the ground 20 times ? In what case , can we consider that it is an infinite series ?

No, this is not an infinite series. Nor is it a geometric series!
First the ball drops 10 m. Then it rebounds (3/4)(10) m. Then it drops (3/4)(10)m, then rebounds m., then drops m. Each distance, after the first drop is doubled once up then down. The total distance is . Notice that it has bounced once after the first drop, twice at the second drop, ... and when the exponent is n, has bounced n+1 times. The total distance traveled at the 20th bounce is
That is NOT a geometric series because the first (n=0) term is 10, not 20. Fortunately, we can fix that by adding 10 to both sides:
Now use the formula .
m, approximately so D= 69.75 m, approximately.
(The total distance after an "infinite" number of bounces would be 70 m.)