Question

4. Find the sum of the following finite series
(i) 1+ 0.1 + 0.01 + 0.001 + .... + (0 1) (ii) 1 +11+ 111+ ..... to 20 terms.

Answer 1

(i)1 + 0.1 + 0.01 + 0.001+ .....
here you observed , 0.1 , 0.01 , 0.001 ,..... Are in GP
where, 0.1 is the first term and 0.1 is the common ratio of it.
Now, sum of infinite term , Sn = a/(1 - r)
= 0.1/(1 - 0.1) = 0.1/0.9 = 1/9
So, 1 + 0.1 + 0.01 + 0.001 + .... = 1 + 1/9 = 10/9

(ii) 1 + 11 + 111 + 1111+ .... 20 terms
= 1/9 [ 9 + 99 + 999 + 9999 + .... 20 terms ]
= 1/9 [(10 - 1) + (10² - 1) + (10³ - 1) + (10⁴ - 1) + ..... 20 terms ]
= 1/9 [ 10 + 10² + 10³ + 10⁴ + .... 20 terms - (1 + 1 + 1 + 1 + ... 20 terms ]
= 1/9 [ 10(10²⁰ - 1)/(10 - 1) - 20 ]
= 1/9 [ 10(10²⁰ - 1)/9 - 20]

Answer 2

GEOMETRIC PROGRESSION :
A sequence a1, a2, a3 ………….. an Is called a geometric sequence if a(n+1) = anr, where r is a non zero constant. Here, a1 is the first term and the constant r is called common ratio. The sequence is also called a geometric progression(G.P)
a1 = a , r = a(ⁿ+1)/ aⁿ

General term of a geometric sequence is
tn = arⁿ-1
General form of G.P is a, a r , a r ²,.........

Sum of n terms in GP ,Sn = a( rⁿ - 1 ) / ( r - 1) [ r> 1] or sn = a(1- rⁿ)⁄(1 - r) [r<1]

Question:
Find Sum of The Following Finite Series
(i) 1+ 0.1 + 0.01 + 0.001 + .... + (0 .1)^9
(ii) 1 +11+ 111+ ..... to 20 terms.

SOLUTION :
GIVEN :
a = 1, r = a(ⁿ+1)/ aⁿ = 0.1/1 = 0.1 , n = 10
sn = a(1- rⁿ)⁄(1 - r)                            [r<1]

S₁₀ = 1 [1- (0.1)^10]/[1- 0.1]
S₁₀ = [1- (0.1)^10]/0.9

(ii)
1 + 11 + 111 + .............. to 20 terms
This series is not a geometric series. To make it a geometric series we have to multiply and divide the whole series by 9.

9/9[1 + 11 + 111 + .............. to 20 terms]

1/9[9 + 99 + 999 + .............. to 20 terms]

1/9[(10-1) + (100-1) + (1000-1) + .............. to 20 terms]

1/9[(10+100+1000+.............. to 20 terms) -(1+1+1+......to 20 terms)]

Here, a = 10 , r = 100/10 = 10

Sn = a(rⁿ-1)⁄(r-1)         [r>1]

Sn= 1/9 [10(10^20 - 1)/(10-1)    - 20 terms]
Sn = 1/9 [10(10^20 - 1)/9    - 20]
Sn = 10/81(10^20 - 1) - 20/9
Sn = 10/81(10^20 - 1) - 20/9

HOPE THIS WILL HELP YOU