# HOW TO FIND THE SUM OF SPECIAL TYPE OF GEOMETRIC SERIES

How to Find the Sum of Special Type of Geometric Series ?

Here we are going to see how to find geometric series with special types.

## How to Find the Sum of Special Type of Geometric Series - Questions

Question 1 :

If the first term of an infinite G.P. is 8 and its sum to infinity is 32/3 then find the common ratio.

Solution :

Since the given sequence is infinite series, the sum of the series  =  a/(1 - r)

a/(1-r)  =  32/3

a = 8

8/(1-r)  =  32/3

8(3)  =  32(1-r)

24 / 32  =  1 - r

3/4  =  1 - r

r  =  1 - (3/4)  ==>  1/4

Question 2 :

Find the sum to n terms of the series

(i) 0.4 + 0.44 + 0.444 +........ to n terms

Solution :

0.4 + 0.44 + 0.444 +........ to n terms

=  4(0.1 + 0.11 + 0.111 + ...... to n terms)

=  4 ⋅ (9/9) (0.1 + 0.11 + 0.111 + ...... to n terms)

=  (4/9)[0.9 + 0.99 + 0.999 +............n terms]

=  (4/9)[(1 - 0.1) + (1 - 0.12) + (1 - 0.13)  + .......n terms]

=  (4/9)[(1+1+1........n terms) - [0.1+0.12+0.13 + .......n terms]

=  (4/9)[n - 0.1 [1 - (0.1)n/(1-0.1)]

=  (4/9)[n - 0.1 [1 - (0.1)n/0.9]

=  (4/9)[n - [(1 - (0.1)n)/0.9]

(ii) 3 + 33 + 333 + ........... to n terms

Solution :

3 + 33 + 333 + ........... to n terms

=  3 [1 + 11 + 111 + ...........+ n terms]

=  3 ⋅ (9/9) (1 + 11 + 111 + ...... to n terms)

=  (3/9)[9 + 99 + 999 +............n terms]

=  (1/3)[(10-1) + (100 - 1) + (1000 - 1) +............n terms]

=(1/3)[(10+100+1000 + .......n terms)-(1+1+1+.........n terms)]

a = 10, r = 100/10  =  10, a = 1 and r = 1

=  (1/3)[10(10n - 1)/(10-1))-n]

=  (1/3)[(10/9)(10n - 1) - n]

=  [(10/27)(10n - 1) - (n/3)]

Question 3 :

Find the sum of the Geometric series

3 + 6 + 12 +............+ 1536

Solution :

tn  =  1536

a  =  3, r = 6/3  =  2

arn-1  =  1536

3(2)n-1  =  1536

2n-1  =  1536/3

2n-1  =  512

2n-1  =  29

n - 1  =  9

n = 10

Now, we have to find the sum of 10 terms.

Sn  =  a(rn - 1)/(r - 1)

S10  =  3(210 - 1)/(2 - 1)

=  3(1024 - 1)

=  3(1023)

S10  =  3069

After having gone through the stuff given above, we hope that the students would have understood, "How to Find the Sum of Special Type of Geometric Series".

Apart from the stuff given in this section "How to Find the Sum of Special Type of Geometric Series"if you need any other stuff in math, please use our google custom search here.