MCQ
$1 + \frac{3}{2} + \frac{7}{4} + \frac{{15}}{8} + \frac{{31}}{{16}} + ...$ $20$ પદ સુધી ... = .....
- A$38 + \frac{1}{{{2^{20}}}}$
- B$39 + \frac{1}{{{2^{19}}}}$
- C$39 + \frac{1}{{{2^{20}}}}$
- D$38 + \frac{1}{{{2^{19}}}}$
$ = \frac{{2 \times {2^r} - 1}}{{{2^r}}}$,
Where $r \ge 0$
$\therefore $ req.sum $ = 1 + \sum\limits_{r = 1}^{19} {\frac{{2 \times {2^r} - 1}}{{{2^r}}}} $
Now, $\sum\limits_{r = 1}^{19} {\left( {\frac{{2 \times {2^r} - 1}}{{{2^r}}}} \right)} = \sum\limits_{r = 1}^{19} {\left( {2 - \frac{1}{{{2^r}}}} \right)} $
$ = 2\left( {19} \right) - \frac{{\frac{1}{2}\left( {1 - {{\left( {\frac{1}{2}} \right)}^{19}}} \right)}}{{1 - \frac{1}{2}}} = 38 + \frac{{{{\left( {\frac{1}{2}} \right)}^{19}} - 1}}{1}$
$ = 38 + {\left( {\frac{1}{2}} \right)^{19}} - 1 = 37 + {\left( {\frac{1}{2}} \right)^{19}}$
$\therefore $ req. sum $ = 1 + 37 + {\left( {\frac{1}{2}} \right)^{19}} = 38 + {\left( {\frac{1}{2}} \right)^{19}}$
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