Question

Compute the sum of first n terms of the following series:
6 + 66 + 666 + 6666 + ...

Answer 1

S_n = 6 + 66 + 666 + ...... n terms

= 6[1 + 11 + 111 + ...... n terms]

= `6/9 [9 + 99 + 999 + ..... "n terms"]`

= `6/9[(10 - 1) + (10^2 - 1) + (10^3 - 1) + ...  "n terms"]`

= `6/9[10 + 10^2 + 10 ^3 .....  "m terms" - (1 + 1 + 1  .... "n terms")]`

= `6/9[(10(10^"n" - 1))/(10 - 1) - "n"]`

= `6/9((10^"n" - 1)/9 - "n")`

= `6/9[(10(10^"n" - 1) - 9"n")/9 - "n"]`

= `6/81[10(10^"n" - 1) - 9"n"]`