**Geometric series :**

A series whose terms are in geometric progression is called geometric series.

To find the sum of n terms of the geometric series, we use one of the formulas given below.

s_{n} = a(r^{n} - 1)/(r - 1) if r > 1

s_{n} = a(1 - r^{n})/(1 - r) if r < 1

s_{n} = a/(1 - r) if r = 1

**Example 1 :**

Find the sum of the series

1 + 3 + 9 + ................. to 10 terms

**Solution :**

a = 1, r = 3 > 1 and n = 10

s_{n} = a(r^{n} - 1)/(r - 1) if r > 1

s_{10} = 1(3^{10} - 1)/(3 - 1)

s_{10} = (59049 - 1)/2

s_{10} = 59048/2

s_{10} = 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?

**Solution :**

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

s_{n} = a / (1 - r)

s_{n} = 6 / (1 - (2/3))

s_{n} = 6 / (1/3)

s_{n} = 18 m

(ii) Distance covered in the upward path

= 4 + 8/3 + 16/9 + ............

Here a = 4 and r = 2/3

s_{n} = a / (1 - r)

s_{n} = 4 / (1 - (2/3))

s_{n} = 4 / (1/3)

s_{n} = 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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