Question

Find the sum of first n terms of the series.   7 77 777 .......

Answer 1

series ,    7 ,  77,  777 ,  7777, .....
  we use the geometric series to be able to do this.

Tn  = 7 * 11111 ... n times
     =  7 * [ 1 + 10 + 100 + ....  + 10^{n-1} ]
     = 7 *  [ 10^n - 1] / [ 10 - 1 ]
     = 7 * [10^n - 1 ] / 9

Sum of first n terms Tn  is  =
  Sn =  Σ (7/9) * 10^n  -  Σ (7/9)
       = (7/9) *  Σ 10^n  - 7 n /9
       = 7/9 *  10 * (10^n -1) / (10 - 1)    -  7 n / 9
         = 70 (10^n -1 ) / 81  - 7 n / 9

           = [ 70 * 10^n - 63 n - 70 ] / 81