hiiiiiiii

 

U have all messed up .... index is organic chemistry .. subject is mathematical induction ...  and actually what is needed is series here ..

 

Never mind ... let me solve this for u ..

 

 

well let S = 7+77+777+...+777_n-digits...7

             

              =7/9 ( 9 + 99 + 999 + .... + 999_n-digits...9)

             

              =7/9 ( [10-1] + [100-1] + ... [10ⁿ - 1] )

             

              = 7/9 ( 10¹ + 10² + 10³ + ... + 10ⁿ - n )

 

              = 7/9 ( 10.[(10ⁿ - 1)/(10-1)] - n )        (Sum of a G.P)

 

              = 7/9 ( [10ⁿ⁺¹ - 10]/9 - n )

  

              = 7/81. ( 10ⁿ⁺¹ - 10 - 9n)

 

 So, we have got the desired answer ......

 

 cheers