1950 AMC 12 Problem 42

Question

The equation

x^{x^{x^{\cdot^{\cdot^{\cdot}}}}}=2

is satisfied when

x

is equal to:

Infinity

2

\sqrt[4]{2}

\sqrt2

None of these

The exponent above the first

x

is the entire infinite tower again

Replace that repeated tower by its given value

2

Let the value of the infinite tower be

T.

Removing its bottom

x

leaves the same tower as the exponent, so

T=x^T.

Since

T=2,

we obtain

2=x^2.

The positive base is therefore

x=\sqrt2,

for which the tower is convergent.

Thus, the correct answer is D.

Problem 42 in other years