MathDB
Finding the Smallest Integer in a Sequence

Source:

March 8, 2009

Problem Statement

Consider a sequence x1,x2,x3,... x_1, x_2, x_3, ..., defined by \begin{align*}x_1 &= \sqrt [3]{3}\\ x_2 &= \sqrt [3]{3} ^ {\sqrt [3]{3}},\end{align*}and in general x_n \equal{} (x_{n \minus{} 1}) ^ {\sqrt [3]{3}}\,\,\text{ for }\,\,n > 1. What is the smallest value of n n for which xn x_n is an integer?
<spanclass=latexbold>(A)</span> 2<spanclass=latexbold>(B)</span> 3<spanclass=latexbold>(C)</span> 4<spanclass=latexbold>(D)</span> 9<spanclass=latexbold>(E)</span> 27 <span class='latex-bold'>(A)</span>\ 2 \qquad <span class='latex-bold'>(B)</span>\ 3 \qquad <span class='latex-bold'>(C)</span>\ 4 \qquad <span class='latex-bold'>(D)</span>\ 9 \qquad <span class='latex-bold'>(E)</span>\ 27