Geometric series :
A series whose terms are in geometric progression is called geometric series.
Example 1 :
Find the sum of the series
1 + 3 + 9 + ................. to 10 terms
a = 1, r = 3 > 1 and n = 10
sn = a(rn - 1)/(r - 1) if r > 1
s10 = 1(310 - 1)/(3 - 1)
s10 = (59049 - 1)/2
s10 = 59048/2
s10 = 29524
Example 2 :
A ball is dropped from a height of 6 m and on each bounce it bounces 2/3 of its previous height.
(i) What is the total length of the downward paths ?
(ii) What is the total length of the upward paths ?
(iii) How far does the ball travel till it stops bouncing?
Distance covered in the downward path
= 6 + 4 + 8/3 + 16/9 + .............
a = 6 and r = 4/6 ==> 2/3
To find the sum of infinite series, we use the formula
sn = a / (1 - r)
sn = 6 / (1 - (2/3))
sn = 6 / (1/3)
sn = 18 m
(ii) Distance covered in the upward path
= 4 + 8/3 + 16/9 + ............
Here a = 4 and r = 2/3
sn = a / (1 - r)
sn = 4 / (1 - (2/3))
sn = 4 / (1/3)
sn = 12 m
(iii) Total distance covered = 18 + 12
= 30 m
After having gone through the stuff given above, we hope that the students would have understood how to find the sum of geometric series.
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