Question

If the interval in which $x(>0)$ must lie so that the greatest term in the expansion of $(1+x{)}^{100}$ has the greatest coefficient in $(a,b)$ then $2ab=$

Answer 1

Greatest coefficient in $(1+x{)}^{100}$
$=$ coefficient of middle term $={}^{100}{C}_{50}$
$\therefore$ Greatest term $={}^{100}{C}_{50}\times x^{50}$

${}^{100}{C}_{50}{x}^{50}>{}^{100}{C}_{49}{x}^{49}\Rightarrow x>\frac{{}^{100}{C}_{49}}{{}^{100}{C}_{50}}$
$\Rightarrow x>\frac{\frac{100!}{49!51!}}{\frac{100!}{50!50!}}\Rightarrow x>\frac{50}{51}$

${}^{100}{C}_{50}{x}^{50}>{}^{100}{C}_{51}{x}^{51}\Rightarrow x<\frac{{}^{100}{C}_{50}}{{}^{100}{C}_{51}}$
$\Rightarrow x<\frac{\frac{100!}{50!50!}}{\frac{100!}{51!49!}}\Rightarrow x>\frac{50}{51}$
$\therefore (a,b)=(\frac{50}{51},\frac{51}{50})$
$\Rightarrow 2ab=2\times \frac{50}{51}\times \frac{51}{50}=2$