1985 AHSME Problems/Problem 27

Problem

Consider a sequence $x_1, x_2, x_3, \cdots$ defined by

$x_1=\sqrt[3]{3}$

$x_2=\sqrt[3]{3}^\sqrt[3]{3}$ (Error compiling LaTeX. Unknown error_msg)

and in general

$x_n=(x_{n-1})^\sqrt[3]{3}$ (Error compiling LaTeX. Unknown error_msg) 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 $x_n=\sqrt[3]{3}^{\left(\sqrt[3]{3}^{n-1}\right)}$

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

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

We now get rid of the cubed roots by introducing fractions into the exponents.

$x_n=\sqrt[3]{3}^{\left(\sqrt[3]{3}^{n-1}\right)}=\sqrt[3]{3}^{\left(3^{\left(\frac{n-1}{3}\right)}\right)}=3^{\left(\frac{1}{3}\cdot3^{\left(\frac{n-1}{3}\right)}\right)}=3^{\left(3^{\left(\frac{n-4}{3}\right)}\right)}$ .

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 $4, \boxed{\text{C}}$ .

1985 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 26	Followed by Problem 28
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1985 AHSME Problems/Problem 27

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