Question

Answer 1

`A = -1/10 - 1/100 - 1/1000 - 1/10000 - 1/100000`

`= -1/10 - 1/(10^2) - 1/(10^3) -1/(10^4) - 1/(10^5) `

`10A = -1 - 1/10 - 1/(10^2) - 1/(10^3) -1/(10^4) `

`10A - A = ( -1 - 1/10 - 1/(10^2) - 1/(10^3) -1/(10^4)) - (-1/10 - 1/(10^2) - 1/(10^3) -1/(10^4) - 1/(10^5) )`

`9A =  -1 - 1/10 - 1/(10^2) - 1/(10^3) -1/(10^4)+1/10 + 1/(10^2) + 1/(10^3) +1/(10^4)+ 1/(10^5) `

`9A = -1 +1/(10^5) `

`9A = (1 - 10^5)/(10^5) `

`A = (1 - 10^5)/(9. 10^5) `

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`B = 1/3 -3/5 +5/7 -7/9 + 9/11 -11/13+13/15 + 11/13 - 9/11 +7/9 - 5/7 + 3/5 `

`= (1/3 + 13/15) + (3/5 - 3/5) + (5/7 - 5/7) + (7/9-7/9) + (9/11 - 9/11) + (11/13 - 11/13)`

`= 5/15 + 13/15 + 0 + 0 + 0 + 0 + 0 `

`= 18/15`

`=6/5`

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`C = 1/99 - 1/(99.98) - 1/(98.97) - ... - 1/(2.1) `

`= 1/99 - (1/(1.2) + 1/(2.3) + ... + 1/(98.99)`

`= 1/99 - (1 - 1/2 + 1/2 - 1/3 +...+ 1/98 - 1/99)`

`= 1/99 - (1  - 1/99)`

`= 1/99 - 1 + 1/99`

`= -97/99`

Answer 2

\(A=-\dfrac{1}{10}-\dfrac{1}{100}-\dfrac{1}{1000}-\dfrac{1}{10000}-\dfrac{1}{100000}\)
\(A=-\left(\dfrac{1}{10}+\dfrac{1}{10^2}+\dfrac{1}{10^3}+\dfrac{1}{10^4}+\dfrac{1}{10^5}\right)\)
\(10A=-\left(\dfrac{10}{10}+\dfrac{10}{10^2}+\dfrac{10}{10^3}+\dfrac{10}{10^4}+\dfrac{10}{10^5}\right)\)
\(10A-A=-\left(1+\dfrac{1}{10}+\dfrac{1}{10^2}+\dfrac{1}{10^3}+\dfrac{1}{10^4}\right)\)
\(9A=-\left(1+\dfrac{1}{10}+\dfrac{1}{10^2}+\dfrac{1}{10^3}+\dfrac{1}{10^4}\right)-\left[-\left(\dfrac{1}{10}-\dfrac{1}{10^2}+\dfrac{1}{10^3}+\dfrac{1}{10^4}+\dfrac{1}{10^5}\right)\right]\)
\(9A=-1-\dfrac{1}{10}-\dfrac{1}{10^2}-\dfrac{1}{10^3}-\dfrac{1}{10^4}+\dfrac{1}{10}+\dfrac{1}{10^2}+\dfrac{1}{10^3}+\dfrac{1}{10^4}+\dfrac{1}{10^5}\)
\(9A=-1+\dfrac{1}{10^5}\)
\(9A=\dfrac{1-10^5}{10^5}\)
\(A=\dfrac{1-10^5}{9.10^5}\)