find-coefficient-of-x-r-in-the-expension-of-2x-6y-8-

Question Number 73628 by Waseem Yaqoob last updated on 14/Nov/19

$${find}\:{coefficient}\:{of}\:{x}^{{r}\:} \:{in}\:{the}\:{expension}\:{of}\: \\ $$$$\:\left(\mathrm{2}{x}−\mathrm{6}{y}\right)^{−\mathrm{8}} \\ $$

Answered by mr W last updated on 14/Nov/19

(2x−6y)^(−8) =(−6y)^(−8) (1−(x/(3y)))^(−8) =(−6y)^(−8) Σ_(r=0) ^∞ C_7 ^(r+7) ((x/(3y)))^r coef. of x^r : a_r =(−6y)^(−8) ((1/(3y)))^r C_7 ^(r+7) =(C_7 ^(r+7) /(2^8 (3y)^(8+r) )) example: a_5 =(C_7 ^(12) /(2^8 (3y)^(13) ))=((11)/(5668704y^(13) ))

$$\left(\mathrm{2}{x}−\mathrm{6}{y}\right)^{−\mathrm{8}} \\ $$$$=\left(−\mathrm{6}{y}\right)^{−\mathrm{8}} \left(\mathrm{1}−\frac{{x}}{\mathrm{3}{y}}\right)^{−\mathrm{8}} \\ $$$$=\left(−\mathrm{6}{y}\right)^{−\mathrm{8}} \underset{{r}=\mathrm{0}} {\overset{\infty} {\sum}}{C}_{\mathrm{7}} ^{{r}+\mathrm{7}} \left(\frac{{x}}{\mathrm{3}{y}}\right)^{{r}} \\ $$$${coef}.\:{of}\:{x}^{{r}} : \\ $$$${a}_{{r}} =\left(−\mathrm{6}{y}\right)^{−\mathrm{8}} \left(\frac{\mathrm{1}}{\mathrm{3}{y}}\right)^{{r}} {C}_{\mathrm{7}} ^{{r}+\mathrm{7}} =\frac{{C}_{\mathrm{7}} ^{{r}+\mathrm{7}} }{\mathrm{2}^{\mathrm{8}} \left(\mathrm{3}{y}\right)^{\mathrm{8}+{r}} } \\ $$$${example}: \\ $$$${a}_{\mathrm{5}} =\frac{{C}_{\mathrm{7}} ^{\mathrm{12}} }{\mathrm{2}^{\mathrm{8}} \left(\mathrm{3}{y}\right)^{\mathrm{13}} }=\frac{\mathrm{11}}{\mathrm{5668704}{y}^{\mathrm{13}} } \\ $$

Commented by Waseem Yaqoob last updated on 16/Nov/19