Sum to 25 terms of the series 0.5 + 0.55 + 0.555 + . . . is:

Let $S = 0.5 + 0.55 + 0.555 + \dots +$ upto $25$ terms

$= \frac{5}{9} [0.9 + 0.99 + 0.999 + \dots +$ up $25$ terms $]$

$\begin{array}{r} = \frac{5}{9} [(1 - 0.1) + (1 - (0.1)^{2}) + (1 - (0.1)^{3}) \\ + \dots + (1 - (0.1)^{25})] \end{array}$

$\begin{array}{l} = \frac{5}{9} [25 - \frac{(0.1) (1 - (0.1)^{25})}{1 - 0.1}] \\ = \frac{5}{81} [224 - 10^{- 25}] \end{array}$

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