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