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