Question-20881

Question Number 20881 by tawa tawa last updated on 05/Sep/17

Answered by mrW1 last updated on 06/Sep/17

(ax+ay)^n =a^n (x+y)^n since a middle term exists, n is even. its middle term is a^n C_(n/2) ^n x^(n/2) y^(n/2) =14580x^3 y^3 ⇒(n/2)=3⇒n=6 C_3 ^6 =20 a^6 ×20=14580 ⇒a=(((14580)/(20)))^(1/6) =3

$$\left(\mathrm{ax}+\mathrm{ay}\right)^{\mathrm{n}} =\mathrm{a}^{\mathrm{n}} \left(\mathrm{x}+\mathrm{y}\right)^{\mathrm{n}} \\ $$$$\mathrm{since}\:\mathrm{a}\:\mathrm{middle}\:\mathrm{term}\:\mathrm{exists},\:\mathrm{n}\:\mathrm{is}\:\mathrm{even}. \\ $$$$\mathrm{its}\:\mathrm{middle}\:\mathrm{term}\:\mathrm{is} \\ $$$$\mathrm{a}^{\mathrm{n}} \mathrm{C}_{\frac{\mathrm{n}}{\mathrm{2}}} ^{\mathrm{n}} \mathrm{x}^{\frac{\mathrm{n}}{\mathrm{2}}} \mathrm{y}^{\frac{\mathrm{n}}{\mathrm{2}}} =\mathrm{14580x}^{\mathrm{3}} \mathrm{y}^{\mathrm{3}} \\ $$$$\Rightarrow\frac{\mathrm{n}}{\mathrm{2}}=\mathrm{3}\Rightarrow\mathrm{n}=\mathrm{6} \\ $$$$\mathrm{C}_{\mathrm{3}} ^{\mathrm{6}} =\mathrm{20} \\ $$$$\mathrm{a}^{\mathrm{6}} ×\mathrm{20}=\mathrm{14580} \\ $$$$\Rightarrow\mathrm{a}=\left(\frac{\mathrm{14580}}{\mathrm{20}}\right)^{\frac{\mathrm{1}}{\mathrm{6}}} =\mathrm{3} \\ $$

Commented by tawa tawa last updated on 08/Sep/17