using mathematical induction

prove that

7+ 77+ 777+ ..............+ 77.........7 = 7/81 (10to the power n+1 -9n-10)

7 + 77 +777 + ……..upto n terms = (-9n -10)

Let , P(n) be the statement given by

P(n) = 7 + 77 +777 + ……..upto n terms = (-9n -10)

Now, we have , P(1) = (-9*1 -10)

  = (100- 19)

  = 7

So, P(1) is true

Now let P(m) is true

 7 + 77 +777 + ……..) = (-9m -10)

We wish to show that P(m+1) is true . for this we have to show that

7 + 77 +777 + ……..(+  = (-9(m+1) -10)

  = (-9m -19)

Now, 7 + 77 +777 + ……..upto m terms+ (m+1) term

= (-9m -10) + )

=  [ + ]

= []

=  + 9. - 9m -10 – 9]

= (1+ 9) - 9m -19]

= .10 - 9m -19]

=  - 9m -19]  

=  - 9m -19]

So, P(m+1) is true

P(m) is true

Also, P(m+1) is true

Hence by the principle of mathematical induction , the given result is true for all n N