AIEEE 2012 (Offline) 29th April Code AQuestion 1
If $n$ is a positive integer, then $\left(\sqrt3+1\right)^{2n}-\left(\sqrt3-1\right)^{2n}$ is :
Binomial Theorem · Integral and Fractional Part of (√a+b)^n
Solution
Concept. In $(a+b)^n-(a-b)^n$ only the odd powers of $b$ survive; here $b=2\sqrt3$. Formulas used. (1) $(a+b)^n-(a-b)^n=2\sum_{k\ \text{odd}}{}^{n}C_k\,a^{n-k}b^k$Step 1 - Square first. $(\sqrt3+1)^2=4+2\sqrt3$ and $(\sqrt3-1)^2=4-2\sqrt3$, so the number is $$(4+2\sqrt3)^n-(4-2\sqrt3)^n.$$ Step 2 - Use (1). Every surviving term has an odd power of $2\sqrt3$, i.e. it is $\sqrt3$ times a positive integer. So the number is $\sqrt3\cdot m$ with $m$ a positive integer. Step 3 - Conclude. $\sqrt3\,m$ is irrational (for example $n=1$ gives $4\sqrt3$). Answer: (D)