Question

The larger of ${99}^{50}+{100}^{50}and{101}^{50}$ is . . .

• A. $(101{)}^{50}$
• B. $(101{)}^{50}=(99{)}^{50}+(100{)}^{50}$
• C. $(99{)}^{50}+(100{)}^{50}$
• D. Can't be determined.

Answer 1

The correct option is A

$(101{)}^{50}$

Consider, $(101{)}^{50}-(99{)}^{50}-(100{)}^{50}$
$=(100+1{)}^{50}-(100-1{)}^{50}-(100{)}^{50}=(100{)}^{50}\left\{\right(1+0.01{)}^{50}-(1-0.01{)}^{50}-1\}=\left(100{)}^{50}\right\{2.{[}^{50}{C}_1\left(0.01\right){+}^{50}{C}_3(0.01{)}^3+\cdots ]-1\}=(100{)}^{50}\left\{2{[}^{50}{C}_3\right(0.01{)}^3{+}^{50}{C}_5(0.01{)}^5+\cdots ]\}\therefore (101{)}^{50}-\left\{\right(99{)}^{50}+\left(100{)}^{50}\right\}>0\Rightarrow (101{)}^{50}>(99{)}^{50}+(100{)}^{50}$