Question

If $n$ is a positive integer, then $\left(\sqrt{3}+1\right)^{2n} - \left(\sqrt{3}-1\right)^{2n}$ is :

• A. an irrational number
• B. an odd positive integer
• C. an even positive integer
• D. a rational number other than positive integers

Answer 1

Answer: (A)

$\left(\sqrt{3} + 1\right)^{2n} -\left(\sqrt{3} - 1\right)^{2n}$
$ = 2 \left[^{2n}C_{1} \left(\sqrt{3}\right)^{2n-1} + ^{2n}C_{3} (\sqrt{3}\right)^{2n-3} + ^{2n} C_{5} \left(\sqrt{3}) ^{2n-5} + ........ \right] $
= which is an irrational number.