7+77+777... n terms sum=??
we take 7/9 as common and get 9,99,999... then we break it in form of 10-1+10*10-1...
is there any other way to solve this sum
s= 7+77+777+..
can we take 7 as common the we get
7(1+11+111...)
therefore we will get the sum of 1+11+111...
as
1*10^(n-1)+2*10^(n-2)...
this is ap gp series so we can solve it
but it is getting tedious
also i dont know whether this is right or wrong
please help
$\displaystyle \displaystyle f(n)=\left(\frac{7}{9}\right)\left(\sum_{i=1}^n(10 ^i-1)\right)$
$\displaystyle \displaystyle =\left(\frac{7}{9}\right)\left(\sum_{i=1}^n10^i-\sum_{i=1}^n1\right)$
$\displaystyle =(\frac{7}{9})((\frac{10}{9})(10^n-1)-n)$
Reducing the first sum in the final step isn't so hard when you notice the numbers are just
10,
110,
1110, ...
To get 1110 you can take 999 and multiply by (10/9).