જો {a_n}, = ,sumlimits_{r, = ,0}^n {frac | vedclass

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Question

$\sum\limits_{r = 0}^n {\,\frac{r}{{\left( {\begin{array}{*{20}{c}}
  n \ 
  r 
\end{array}} ight)}}} \, = \,\frac{0}{{\left

Vedclass · Accepted Answer

$\frac{n}{2}\,{a_n}$

Vedclass · Answer

$\sum\limits_{r = 0}^n {\,\frac{r}{{\left( {\begin{array}{*{20}{c}}
  n \ 
  r 
\end{array}} ight)}}} \, = \,\frac{0}{{\left( {\begin{array}{*{20}{c}}
  n \ 
  0 
\end{array}} ight)}}\, + \,\frac{1}{{\left( {\begin{array}{*{20}{c}}
  n \ 
  1 
\end{array}} ight)}}\, + \,\frac{2}{{\left( {\begin{array}{*{20}{c}}
  n \ 
  2 
\end{array}} ight)}}\, + \,.....\, + \,\frac{{n\, - \,1}}{{\left( {\begin{array}{*{20}{c}}
  n \ 
  {n\, - \,1} 
\end{array}} ight)}}\, + \,\frac{n}{{\left( {\begin{array}{*{20}{c}}
  n \ 
  n 
\end{array}} ight)}}$

$\sum\limits_{r = 0}^n {\,\frac{r}{{\left( {\begin{array}{*{20}{c}}
  n \ 
  r 
\end{array}} ight)}}} \, = \,\frac{0}{{\left( {\begin{array}{*{20}{c}}
  n \ 
  n 
\end{array}} ight)}}\, + \,\frac{1}{{\left( {\begin{array}{*{20}{c}}
  n \ 
  {n\, - \,1} 
\end{array}} ight)}}\, + \,\frac{2}{{\left( {\begin{array}{*{20}{c}}
  n \ 
  {n\, - \,2} 
\end{array}} ight)}}\, + \,......\, + \,\frac{{n\, - \,1}}{{\left( {\begin{array}{*{20}{c}}
  n \ 
  1 
\end{array}} ight)}}\, + \,\frac{n}{{\left( {\begin{array}{*{20}{c}}
  n \ 
  0 
\end{array}} ight)}}\,....\,(2)\,$

$\left[ {\because \,\left( {\begin{array}{*{20}{c}}
  n \ 
  r 
\end{array}} ight)\, = \,\left( {\begin{array}{*{20}{c}}
  n \ 
  {n\, - \,2} 
\end{array}} ight)} ight]$

હવે 

$\,{	ext{(1)  +  (2) }}\, \Rightarrow \,2\,\sum\limits_{r\, = \,0}^n {\frac{r}{{\left( {\begin{array}{*{20}{c}}
  n \ 
  r 
\end{array}} ight)}}\, = \,\frac{n}{{\left( {\begin{array}{*{20}{c}}
  n \ 
  0 
\end{array}} ight)}}} \, + \,\frac{n}{{\left( {\begin{array}{*{20}{c}}
  n \ 
  1 
\end{array}} ight)}}\, + \,......\, + \,\frac{n}{{\left( {\begin{array}{*{20}{c}}
  n \ 
  n 
\end{array}} ight)}}$

$ = \,n\,\left[ {\frac{1}{{\left( {\begin{array}{*{20}{c}}
  n \ 
  0 
\end{array}} ight)}}\, + \,\frac{1}{{\left( {\begin{array}{*{20}{c}}
  n \ 
  1 
\end{array}} ight)}}\, + \,.....\, + \,\frac{1}{{\left( {\begin{array}{*{20}{c}}
  n \ 
  n 
\end{array}} ight)}}} ight] = \,n\,\sum\limits_{r\, = \,0}^n {\frac{1}{{\left( {\begin{array}{*{20}{c}}
  n \ 
  r 
\end{array}} ight)}}} \, = \,n{a_n}$

$\sum\limits_{r\, = 0}^n {\,\frac{r}{{\left( {\begin{array}{*{20}{c}}
  n \ 
  r 
\end{array}} ight)}}} \, = \,\frac{n}{2}\,{a_n}$

જો ${a_n}\, = \,\sum\limits_{r\, = \,0}^n {\frac{1}{{\left( {\begin{array}{{20}{c}}
n \\
r
\end{array}} \right)}}} $ તો $\sum\limits_{r\, = \,0}^n {\,\frac{r}{{\left( {\begin{array}{{20}{c}}
n \\
r
\end{array}} \right)}}\, = \,.....} $

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