Question

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

• A. $652x^2,\frac{5436}{x^4}$
• B. $672x^3,\frac{5376}{x^3}$
• C. $672x^4,\frac{536}{x^2}$
• D. None of these

Answer 1

The correct option is B $672x^3,\frac{5376}{x^3}$

given expression is ${(x+\frac{2}{x})}^9$

To find $4^{th}$ term from beginning, then

$T_4=T_{3+1}{=}^9{C}_3(x{)}^{9-3}{\left(\frac{2}{x}\right)}^3$

$=\frac{9!}{6!3!}\left(x^6\right)\left(\frac{8}{x^3}\right)$

$=\frac{9\times 8\times 7\times 6!}{6!\times 3\times 2}\times 8\times x^3$

$T_4=672x^3$

To find term from the end, then

$T_{9-P+2}=T_{9-4+2}=T_7$

$T_7{=}^9{C}_3(x{)}^{9-6}{\left(\frac{2}{x}\right)}^6$

$=\frac{9!}{6!3!}{x}^3\frac{2^6}{x^6}$

$=\frac{9\times 8\times 7\times 6!}{6!\times 3\times 2}\times \frac{2^6}{x^3}$

$T_7=\frac{5376}{x^3}$