Question

Find the sum of the following series to n terms :-

0.5+0.55+0.555+.........

Using simplest method

Answer 1

Let
S=0.5+0.55+0.555+.....n terms
Taking 5 as common we get
S=5(0.1+0.11+0.111+...nterms)
Multiply rhs with 9/9
S=(5×9/9)[0.1+0.11+0.111+.....n terms]
S=5/9[0.9+0.99+0.999+....n terms]
S=5/9[(1-0.1)+(1-0.01)+(1-0.001)+....n terms]
S=5/9[(1+1+1+1+...n terms)-(0.1+0.01+0.001+....n terms)]----(1)
Now find sum of each bracket. Separately
i)1+1+1+...n terms =n----(2)
ii)0.1+0.01+0.001+....n terms
Here
First term =a = 0.1=1/10
Common ratio= r= a2/a1=0.01/0.1 =0.1=1/10
r<1
Therefore the series is in G P.
Sum =[a(1-r^n)]/(1-r)

=[1/10(1-1/10^n)]/(1-1/10)
=[1/10(1-1/10^n)]/(9/10)
=(1-1/10^n)/9----(3)
Put (2) and (3) in (1)
S=5/9[n -(1-1/10^n)/9]
S=5n/9- 5/81(1-1/10^n)