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(1) + 6(11) + 6(111) + …to n terms

taking 6 common

= 6[1 + 11 + 111 + …to n terms]

divide & multiply by 9

= (6/9)[9(1 + 11 + 111 + …to n terms)]

= (6/9)[9 + 99 + 999 + …to n terms]

= (6/9)[(10 - 1) + (100 - 1) + (1000 - 1) + …to n terms]

= (6/9)[(10 - 1) + (10² - 1) + (10³ - 1) + …to n terms]

= (6/9)[{10 + 10² + 10³ + …n terms} - {1 + 1 + 1 + …n terms}]

= (6/9)[{10 + 10² + 10³ + …n terms} - n]

Since 10 + 10² + 10³ + …n terms is in GP with

first term(a) = 10

common ratio(r) = 10²/10 = 10

We know that

Sum of n terms = (As r>1)

putting value of a & r

10 + 10² + 10³ + …n terms

Hence, Sum