1985 AMC 12 Problem 27

Question

Consider a sequence

x_1,

x_2,

x_3,

\ldots,

defined by

\begin{aligned} x_1&=\sqrt[3]{3},\\ x_2&=(\sqrt[3]{3})^{\sqrt[3]{3}}. \end{aligned}

and in general

x_n=(x_{n-1})^{\sqrt[3]{3}}

for

n\gt1.

What is the smallest value of

n

for which

x_n

is an integer?

2

3

4

9

27

Let

c=\sqrt[3]{3}

and track the exponent of

3

Show that

x_n=3^{\frac{c^{\,n-1}}{3}}

and examine the first four terms

Let

c=\sqrt[3]{3}.

Repeated application of the recurrence gives

x_n=3^{\frac{c^{\,n-1}}{3}},

so

x_4=3^{\frac{c^3}{3}}=3.

It remains to rule out the first three terms. The sequence is strictly increasing. Also

c\lt\frac{3}{2},

so

x_2=c^c \lt\left(\frac32\right)^{\frac{3}{2}} \lt2.

Finally

x_3=3^{\frac{1}{c}}.

Since

2^c\lt2^{\frac{3}{2}}\lt3,

we have

x_3\gt2,

while

\frac{1}{c}\lt1

gives

x_3\lt3.

Thus

x_1,x_2,x_3

are not integers, and the first integral term is

x_4.

Therefore the correct answer is C.

Problem 27 in other years