Question

If $a={51}^{25}$ and $b={49}^{25}+{50}^{25}$, then which of the following option is correct?

• A. $a=b$
• B. $a>b$
• C. $b>a$
• D. $a-b$ is an odd integer

Answer 1

The correct option is B $a>b$

Given : $a={51}^{25}$ and $b={49}^{25}+{50}^{25}$
Now,
${51}^{25}-{49}^{25}=(50+1{)}^{25}-(50-1{)}^{25}=2\left[{}^{25}{C}_1\right(50{)}^{24}+{}^{25}{C}_3(50{)}^{22}+\cdots ]=\left[\right(50{)}^{25}+2\cdot {}^{25}{C}_3(50{)}^{22}+\cdots ]$
Therefore,
${51}^{25}-{49}^{25}-(50{)}^{25}=2\cdot {}^{25}{C}_3(50{)}^{22}+\cdots \Rightarrow {51}^{25}-{49}^{25}-{50}^{25}>0$
Clearly, $a-b$ is an even integer.
$a>b$