Question

$\small{\displaystyle \tt \red{ \frac{1}{ {10}^{ - 9} + 1 } + \frac{1}{ {10}^{ - 8} + 1} + \cdot \cdot \cdot \frac{1}{ {10}^{8} + 1} + \frac{1}{ {10}^{9} + 1}}}$​

Answer 1

$\textsf{\large{\underline{Solution}:}}$

Given Expression:

$\rm = \dfrac{1}{ {10}^{ - 9} + 1 } + \dfrac{1}{ {10}^{ - 8} + 1} + \cdot \cdot \cdot \dfrac{1}{ {10}^{8} + 1} + \dfrac{1}{ {10}^{9} + 1}$

Now consider the first term.

$\rm = \dfrac{1}{ {10}^{ - 9} + 1 }$

$\rm = \dfrac{1}{ \dfrac{1}{ {10}^{9} } + 1 }$

$\rm = \dfrac{1}{ \dfrac{ {10}^{9} + 1}{ {10}^{9} }}$

$\rm = \dfrac{ {10}^{9} }{ {10}^{9} + 1}$

You will get similar result for others too.

Therefore, it becomes:

$\rm = \dfrac{ {10}^{9} }{ {10}^{9} + 1 } + \dfrac{ {10}^{8} }{ {10}^{8} + 1} + \cdot \cdot \cdot \dfrac{1}{ {10}^{8} + 1} + \dfrac{1}{ {10}^{9} + 1}$

$\rm = \dfrac{ {10}^{9} + 1}{ {10}^{9} + 1 } + \dfrac{ {10}^{8} + 1}{ {10}^{8} + 1} + \cdot \cdot \cdot \dfrac{ {10}^{1} + 1}{ {10}^{1} + 1} + \dfrac{1}{ {10}^{0} + 1}$

$\rm = 1 + 1 + .. \: 9 \: times + \dfrac{1}{1+ 1}$

$\rm = 9 + \dfrac{1}{2}$

$\rm = 9\dfrac{1}{2}$

Which is our required answer.

$\textsf{\large{\underline{Answer}:}}$

$\rm\hookrightarrow\dfrac{1}{ {10}^{ - 9} + 1 } + \dfrac{1}{ {10}^{ - 8} + 1} + \cdot \cdot \cdot \dfrac{1}{ {10}^{8} + 1} + \dfrac{1}{ {10}^{9} + 1} = 9 \dfrac{1}{2}$