Question

Sum upto $n$ terms the series

$S=6+66+666+.......$ be $\frac{6}{m}[{10}^{n+1}-kn-10]$.Find $m/k$ ?

Answer 1

$S=6+66+666+...$

$=6+60+6+600+60+6+...$

$=6+6\left(10\right)+6+6\left({10}^2\right)+6\left(10\right)+6+...$

$=6[n+(n-1)10+(n-2){10}^2+...{10}^{n-1}]$

$\Rightarrow 10S=6[n10+(n-1){10}^2+...10^n]$

$\Rightarrow (1-10)S=6[n-10-{10}^2...-{10}^{n-1}+{10}^n]$

$\Rightarrow S=\frac{6}{63}[{10}^{n+1}-7n-10]$

$\therefore m=63,k=7$

$\Rightarrow \frac{63}{7}=9$