Question

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

• A. ${99}^{50}+{100}^{50}$
• B. ${101}^{50}$
• C. Both are equal.
• D. None of the above.

Answer 1

Answer: (B)

${101}^{50}$

${(x-1)}^n={{}^{n}C}_o{x}^n-{{}^{n}C}_1{x}^{n-1}+{{}^{n}C}_2{x}^{n-2}+......(-1{)}^n{{}^{n}C}_n.$

${(x+1)}^n={{}^{n}C}_o{x}^n+{{}^{n}C}_1{x}^{n-1}+{{}^{n}C}_2{x}^{n-2}+......{{}^{n}C}_n.$

${(x-1)}^n+x^n$ is less than ${(x+1)}^n;$we can clearly see that, and the difference would further increase for greater $x$ &$n$

Take, $x=100$ and $n=50$

$={99}^{50}+{100}^{50}<{101}^{50}$.

Hence the answer is ${101}^{50}.$