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-

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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 .

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

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