# Difference between revisions of "1985 AHSME Problems/Problem 27"

## Problem

Consider a sequence $x_1, x_2, x_3, \cdots$ defined by $x_1=\sqrt{3}$

$x_2=\sqrt{3}^\sqrt{3}$ (Error compiling LaTeX. ! Missing { inserted.)

and in general

$x_n=(x_{n-1})^\sqrt{3}$ (Error compiling LaTeX. ! Missing { inserted.) for $n>1$.

What is the smallest value of $n$ for which $x_n$ is an integer? $\mathrm{(A)\ } 2 \qquad \mathrm{(B) \ }3 \qquad \mathrm{(C) \ } 4 \qquad \mathrm{(D) \ } 9 \qquad \mathrm{(E) \ }27$

## Solution

First, we will use induction to prove that $x_n=\sqrt{3}^{\left(\sqrt{3}^{n-1}\right)}$

We see that $x_1=\sqrt{3}=\sqrt{3}^{\left(\sqrt{3}^0\right)}$. This is our base case.

Now, we have $x_n=(x_{n-1})^{\sqrt{3}}=\left(\sqrt{3}^{\left(\sqrt{3}^{n-2}\right)}\right)^{\sqrt{3}}=\sqrt{3}^{\left(\sqrt{3}\cdot\sqrt{3}^{n-2}\right)}=\sqrt{3}^{\left(\sqrt{3}^{n-1}\right)}$. Thus the induction is complete.

We now get rid of the cubed roots by introducing fractions into the exponents. $x_n=\sqrt{3}^{\left(\sqrt{3}^{n-1}\right)}=\sqrt{3}^{\left(3^{\left(\frac{n-1}{3}\right)}\right)}=3^{\left(\frac{1}{3}\cdot3^{\left(\frac{n-1}{3}\right)}\right)}=3^{\left(3^{\left(\frac{n-4}{3}\right)}\right)}$.

Notice that since $3$ isn't a perfect power, $x_n$ is integral if and only if the exponent, $3^{\left(\frac{n-4}{3}\right)}$, is integral. By the same logic, this is integeral if and only if $\frac{n-4}{3}$ is integral. We can now clearly see that the smallest positive value of $n$ for which this is integral is $4, \boxed{\text{C}}$.

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 