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

6 + . 66 + . 666 + …

The given sum is not in GP but we can write it as follows: -

Sum = .6 + .66 + .666 + …to n terms

= 6(0.1) + 6(0.11) + 6(0.111) + …to n terms

taking 6 common

= 6[0.1 + 0.11 + 0.111 + …to n terms]

divide & multiply by 9

= (6/9)[9(0.1 + 0.11 + 0.111 + …to n terms)]

= (6/9)[0.9 + 0.99 + 0.999 + …to n terms]

Since is in GP with

first term(a) = 1/10

common ratio(r) = 10 ^{- 2}/10 ^{- 1} = 10 ^{- 1} = 1/10

We know that

Sum of n terms = (As r<1)

putting value of a & r

Hence, Sum