Question and Answers Forum

All Questions      Topic List

UNKNOWN Questions

Previous in All Question      Next in All Question      

Previous in UNKNOWN      Next in UNKNOWN      

Question Number 18168 by allizzwell last updated on 16/Jul/17

If the coefficient of the middle term in  the expansion of the (1+x)^(2n+2)  is p and  the coefficients of middle terms in the  expansion of (1+x)^(2n+1)  are q and r, then

$$\mathrm{If}\:\mathrm{the}\:\mathrm{coefficient}\:\mathrm{of}\:\mathrm{the}\:\mathrm{middle}\:\mathrm{term}\:\mathrm{in} \\ $$$$\mathrm{the}\:\mathrm{expansion}\:\mathrm{of}\:\mathrm{the}\:\left(\mathrm{1}+{x}\right)^{\mathrm{2}{n}+\mathrm{2}} \:\mathrm{is}\:{p}\:\mathrm{and} \\ $$$$\mathrm{the}\:\mathrm{coefficients}\:\mathrm{of}\:\mathrm{middle}\:\mathrm{terms}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{expansion}\:\mathrm{of}\:\left(\mathrm{1}+{x}\right)^{\mathrm{2}{n}+\mathrm{1}} \:\mathrm{are}\:{q}\:\mathrm{and}\:{r},\:\mathrm{then} \\ $$

Answered by ajfour last updated on 16/Jul/17

p=^(2n+2) C_(n+1)   q=^(2n+1) C_n    =  r=^(2n+1) C_(n+1)   further ^(2n+2) C_(n+1) =((2n+2)/(n+1))^(2n+1) C_n   ⇒     p=2q=2r .

$$\mathrm{p}=^{\mathrm{2n}+\mathrm{2}} \mathrm{C}_{\mathrm{n}+\mathrm{1}} \\ $$$$\mathrm{q}=^{\mathrm{2n}+\mathrm{1}} \mathrm{C}_{\mathrm{n}} \:\:\:=\:\:\mathrm{r}=^{\mathrm{2n}+\mathrm{1}} \mathrm{C}_{\mathrm{n}+\mathrm{1}} \\ $$$$\mathrm{further}\:\:^{\mathrm{2n}+\mathrm{2}} \mathrm{C}_{\mathrm{n}+\mathrm{1}} =\frac{\mathrm{2n}+\mathrm{2}}{\mathrm{n}+\mathrm{1}}\:^{\mathrm{2n}+\mathrm{1}} \mathrm{C}_{\mathrm{n}} \\ $$$$\Rightarrow\:\:\:\:\:\mathrm{p}=\mathrm{2q}=\mathrm{2r}\:. \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com