Menu Close

if-a-1-a-2-a-3-a-4-are-the-coefficient-of-any-four-four-consecutive-terms-in-the-expansion-of-1-x-n-then-a-1-a-2-a-1-a-3-a-3-a-4-is-equal-to-




Question Number 63858 by gunawan last updated on 10/Jul/19
if a_1 , a_2 , a_3 , a_4  are the coefficient  of any four four consecutive  terms in the expansion of (1+x)^n   then (a_1 /(a_2 +a_1 ))+(a_3 /(a_3 +a_4 )) is equal to...
$$\mathrm{if}\:\mathrm{a}_{\mathrm{1}} ,\:\mathrm{a}_{\mathrm{2}} ,\:\mathrm{a}_{\mathrm{3}} ,\:\mathrm{a}_{\mathrm{4}} \:\mathrm{are}\:\mathrm{the}\:\mathrm{coefficient} \\ $$$$\mathrm{of}\:\mathrm{any}\:\mathrm{four}\:\mathrm{four}\:\mathrm{consecutive} \\ $$$$\mathrm{terms}\:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of}\:\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{n}} \\ $$$$\mathrm{then}\:\frac{\mathrm{a}_{\mathrm{1}} }{\mathrm{a}_{\mathrm{2}} +\mathrm{a}_{\mathrm{1}} }+\frac{\mathrm{a}_{\mathrm{3}} }{\mathrm{a}_{\mathrm{3}} +\mathrm{a}_{\mathrm{4}} }\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to}… \\ $$
Answered by ajfour last updated on 10/Jul/19
v=((^n C_(r−1) )/(^n C_r +^n C_(r−1) ))+((^n C_(r+1) )/(^n C_(r+1) +^n C_(r+2) ))  &  ^n C_r +^n C_(r−1) =^(n+1) C_r   ⇒  v=Σ_(r=r) ^(r+1) ((n!r!(n−r+1)!)/((r−1)!(n−r+1)!(n+1)!))     =(r/(n+1))+((r+1)/(n+1))=((2r+1)/(n+1)) .
$${v}=\frac{\:^{{n}} {C}_{{r}−\mathrm{1}} }{\:^{{n}} {C}_{{r}} +\:^{{n}} {C}_{{r}−\mathrm{1}} }+\frac{\:^{{n}} {C}_{{r}+\mathrm{1}} }{\:^{{n}} {C}_{{r}+\mathrm{1}} +\:^{{n}} {C}_{{r}+\mathrm{2}} } \\ $$$$\&\:\:\:^{{n}} {C}_{{r}} +\:^{{n}} {C}_{{r}−\mathrm{1}} =\:^{{n}+\mathrm{1}} {C}_{{r}} \\ $$$$\Rightarrow \\ $$$${v}=\underset{{r}={r}} {\overset{{r}+\mathrm{1}} {\sum}}\frac{{n}!{r}!\left({n}−{r}+\mathrm{1}\right)!}{\left({r}−\mathrm{1}\right)!\left({n}−{r}+\mathrm{1}\right)!\left({n}+\mathrm{1}\right)!} \\ $$$$\:\:\:=\frac{{r}}{{n}+\mathrm{1}}+\frac{{r}+\mathrm{1}}{{n}+\mathrm{1}}=\frac{\mathrm{2}{r}+\mathrm{1}}{{n}+\mathrm{1}}\:. \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *