Question

If $x=99^{50}+100^{50}$ and $y={\left(101\right)}^{50}$ then

• A. $x=y$
• B. $x<y$
• C. $x>y$
• D. None of these

Answer 1

Answer: (B)

${\left(101\right)}^{50}-{\left(99\right)}^{50}={\left(100+1\right)}^{50}-{\left(100-1\right)}^{50}$
$=2\left[{}^{50}{C}_1{\left(100\right)}^{49}{+}^{50}{C}_3{\left(100\right)}^{47}+......{+}^{50}{C}_{49}\left(100\right)\right]$
$>2{.}^{50}{C}_1.{\left(100\right)}^{49}=2\times 50{\left(100\right)}^{49}={\left(100\right)}^{50}$
$\Rightarrow {\left(101\right)}^{50}>{\left(99\right)}^{50}+{\left(100\right)}^{50}\Rightarrow y>x\Rightarrow x<y.$