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.