Question

$0.5+0.55+0.555+$________ to $25$ terms.

Answer 1

Let $S=0.5+0.55+0.555+...$ to 25 terms

$S=0.5+0.55+0.555+...+\mathrm{0.55...}\left({255}^{\prime }S\right)$

$0.15=0.05+0.055+....+\mathrm{0.055...}\left({245}^{\prime }S\right)$

$+\mathrm{0.0555...}\left({255}^{\prime }S\right)$

$0.95=0.5+0.5+0.5+...+0.5$

$+\mathrm{0.0555...}\left({255}^{\prime }S\right)$

$\Rightarrow 0.9S=25\times 0.5+\mathrm{0.0555...}\left({255}^{\prime }S\right)$

$\Rightarrow S=\frac{12.5}{0.9}+\frac{\mathrm{0.0555...}\left({255}^{\prime }S\right)}{0.9}$