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Question Number 106690 by  M±th+et+s last updated on 06/Aug/20
this question was repeatd six times  in a various exams between 1971 to 2001.  if C_0 ,C_1 ,C_2 .......,C_n  are the coefficients  in the expansion of (1+x)^n  then  c_0 +2C_1 +3C_2 ........(n+1)C_n =?
$${this}\:{question}\:{was}\:{repeatd}\:{six}\:{times} \\ $$$${in}\:{a}\:{various}\:{exams}\:{between}\:\mathrm{1971}\:{to}\:\mathrm{2001}. \\ $$$${if}\:{C}_{\mathrm{0}} ,{C}_{\mathrm{1}} ,{C}_{\mathrm{2}} …….,{C}_{{n}} \:{are}\:{the}\:{coefficients} \\ $$$${in}\:{the}\:{expansion}\:{of}\:\left(\mathrm{1}+{x}\right)^{{n}} \:{then} \\ $$$${c}_{\mathrm{0}} +\mathrm{2}{C}_{\mathrm{1}} +\mathrm{3}{C}_{\mathrm{2}} ……..\left({n}+\mathrm{1}\right){C}_{{n}} =? \\ $$
Answered by prakash jain last updated on 06/Aug/20
(r+1)C_r =rC_r +C_r   (1+x)^n =Σ_(j=0) ^n C_j x^j   differentiate  n(1+x)^(n−1) =Σ_(j=0) ^n jC_j x^(j−1)   x=1  n2^(n−1) =Σ_(j=0) ^n jC_j   Σ_(r=0) ^n (r+1)C_r =Σ_(r=0) ^n rC_r +Σ_(r=0) ^n C_r   =n2^(n−1) +2^n
$$\left({r}+\mathrm{1}\right){C}_{{r}} ={rC}_{{r}} +{C}_{{r}} \\ $$$$\left(\mathrm{1}+{x}\right)^{{n}} =\underset{{j}=\mathrm{0}} {\overset{{n}} {\sum}}{C}_{{j}} {x}^{{j}} \\ $$$${differentiate} \\ $$$${n}\left(\mathrm{1}+{x}\right)^{{n}−\mathrm{1}} =\underset{{j}=\mathrm{0}} {\overset{{n}} {\sum}}{jC}_{{j}} {x}^{{j}−\mathrm{1}} \\ $$$${x}=\mathrm{1} \\ $$$${n}\mathrm{2}^{{n}−\mathrm{1}} =\underset{{j}=\mathrm{0}} {\overset{{n}} {\sum}}{jC}_{{j}} \\ $$$$\underset{{r}=\mathrm{0}} {\overset{{n}} {\sum}}\left({r}+\mathrm{1}\right){C}_{{r}} =\underset{{r}=\mathrm{0}} {\overset{{n}} {\sum}}{rC}_{{r}} +\underset{{r}=\mathrm{0}} {\overset{{n}} {\sum}}{C}_{{r}} \\ $$$$={n}\mathrm{2}^{{n}−\mathrm{1}} +\mathrm{2}^{{n}} \\ $$
Commented by  M±th+et+s last updated on 06/Aug/20
well done sir
$${well}\:{done}\:{sir} \\ $$

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