Question

Find the 4th term from the beginning and 4th term from the end in the expansion of ${(x+\frac{2}{x})}^9$

Answer 1

Term from the beginning

$T_N=T_{r+1}={}^n{C}_r{x}^{n-r}{y}^r....\left(i\right)$

$N=4,r=3,n=9,x=x,y=\frac{2}{x}$

$T_4=T_{3+1}={}^9{C}_3{x}^6{\left(\frac{2}{x}\right)}^3=\frac{9\times 7\times 8}{3\times 2}{x}^3{x}^8=672x^3$

$4^{th}$ term from the end $=7^{th}$ term from beginning Using (i)

$N=7,r=6,n=9,x=x,y=\frac{2}{x}$

$T_7=T_{6+1}={}^9{C}_6{x}^3{\left(\frac{2}{x}\right)}^6=\frac{9\times 8\times 7}{3\times 2}\times \frac{2^6}{x^3}$

$=\frac{5376}{x^3}$