6+66+666+ n terms - Maths - Principle of Mathematical Induction

Question

6+66+666+ n terms = 2/7 (10 power of n+1 - 9n -10)
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Akshay Chinchole · Answer

we first we need to find a way to represent 666 which is a number
with n 6's
this is simply equal to 6*sum(10^i,i=0 to n-1)
sum(10^i,i=0 to n-1) is simply a geometric summation and is equal
to
[(10^n)-1]/9
and thus the number with n 6's is given by
2*[(10^n)-1]/3
so now we want to take the sum of
2*[(10^i)-1]/3 for i=1 to n
now you can split this into 2 sums
2*(10^i)/3 and -2/3, the first is a geometric summation and the
second is equal to -2n/3
the first is equal to
2*[10^(n+1)-10]/27
thus the total summation is equal to
2*[10^(n+1)-10]/27 - (2n/3)
[2*10^(n+1)-20-18n]/27

6+66+666+..................n terms = 2/7 (10 power of n+1 - 9n -10) =

6+66+666+..................n terms = 2/7 (10 power of n+1 - 9n -10)

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