The sum \sum_{k=1}^{20}k\frac{1}{2^{k}} is equal to :
 

  • Option 1)

    2-\frac{3}{2^{17}}

  • Option 2)

    1-\frac{11}{2^{20}}

  • Option 3)

    2-\frac{11}{2^{19}}

     

  • Option 4)

    2-\frac{21}{2^{20}}

 

Answers (1)

S=\sum_{k=1}^{20}\frac{k}{2^{k}}=\frac{1}{2}+\frac{2}{2^{2}}+\frac{3}{2^{3}}+\cdots \frac{20}{2^{20}}\cdots (1)

Multiply \frac{1}{2} both side

\frac{1}{2}S=\frac{1}{2^{2}}+\frac{2}{2^{3}}+\frac{3}{2^{4}}+\cdots \frac{20}{2^{21}}\cdots (2)

Substract (1) - (2)

\frac{1}{2}S=(\frac{1}{2}+\frac{1}{2^{2}}+\frac{1}{2^{3}}+\cdots \frac{1}{2^{20}})-\frac{20}{2^{21}}

\frac{1}{2}S=\frac{\frac{1}{2}\left ( 1-(\frac{1}{2})^{20} \right )}{1-\frac{1}{2}}-\frac{20}{2^{21}}

\frac{1}{2}S=1-\frac{1}{2^{20}}-\frac{20}{2^{21}}=1-\frac{11}{2^{20}}

S=2-\frac{11}{2^{19}}


Option 1)

2-\frac{3}{2^{17}}

Option 2)

1-\frac{11}{2^{20}}

Option 3)

2-\frac{11}{2^{19}}

 

Option 4)

2-\frac{21}{2^{20}}

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