Question
Find the summation for the series-: $$5+55+555+5555+...$$
I know it is a duplicate of this, but still, I am posting this because i was thinking of solving another way due to which I got stuck in another series.
My Attempt
$S=5+55+555+5555....$
$2*S=10+110+1110+11110....$
$=10*(1+11+111+1111+...)$
$=\frac{10}{9}*(9+99+999+9999+...)$
$=\frac{10}{9}*((10-1)+(10^{2}-1)+(10^{3}-1)+(10^{4}-1)+...)$
$=\frac{10}{9}*(10*\frac{10^{n}-1}{10-1}-n)$
$=\frac{10}{9}*(\frac{10^{n+1}-10}{9}-n)$
$$S=\frac{5}{9}*(\frac{10^{n+1}-10}{9}-n)$$
So i think i got my answer.But i have doubt in summation of series-:
$S=10+110+1110+11110....$
I got summation (from above ) as-: $=\frac{10}{9}*(\frac{10^{n+1}-10}{9}-n)$
But here , it is different .Which one is correct? I am stuck . Please help me out !