Question

Greatest term in the binomial expansion of $(a+2x{)}^9$ when $a=1\&x=\frac{1}{3}$ is :

• A. $3^{\text{rd }}\&4^{\text{th }}$
• B. $4^{\text{th }}\&5^{\text{th }}$
• C. only $4^{\text{th }}$
• D. only $5^{\text{th }}$

Answer 1

Answer: (B)

$\frac{T_{r+1}}{T_r}=\frac{{}^9{C}_r{a}^{9-r}\cdot (2x{)}^r}{{}^9{C}_{r-1}{a}^{9-r+1}(2x{)}^{r-1}}=\frac{9-r+1}{r}\frac{2x}{a}=\frac{9-r+1}{r}\frac{2}{3}=\frac{10-r}{r}\frac{2}{3}$
$\therefore T_{r+1}>T_r$ if $\frac{10-r}{r}\frac{2}{3}>1$ i.e. $20>5r$ i.e. $r<4$
Hence for values of ' $r$ ' upto 3, $T_{r+1}<T_r$
when $r=4,T_{r+1}=T_r$ i.e. $T_4=T_5\&$ both of them are the greatest terms