The ball rolled for a distance of 32 cm in the first second, 16 cm after the second seconds, 8 cm in the third seconds and so on. What is the distance that the ball traveled before it stops?
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3 2 , 1 6 , 8 , . . . ⟹ Observed that it is a geometric progression with a common ratio of 1 6 8 = 3 2 1 6 = 0 . 5
In here we are finding the sum of the geometric progression with infinitely many terms. If S represents this sum then
S = 1 − r a 1 where a 1 is the first term and r is the common ratio. Substituting, we have
S = 1 − 0 . 5 3 2 = 6 4
Here we see that the terms for distance are decreasing by 2.
They will continue to decrease from 2 1 to 4 1 and so on.
Therefore it is an infinite geometric progression.
So a = 32 and r = 2 1
Therefore S n = 1 − r a = 0 . 5 3 2 = 6 4
So the answer is 6 4
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Using the Infinite Geometric Series, a 1 = 32 and the common ratio is 2 1
Simplifying, we have S n = 1 − r a 1 = 1 − 2 1 3 2 = 64
Hence the answer is 6 4