1951 AMC 12 Problem 42

Attempt Problem 42 of the 1951 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1951 AMC 12 solutions, or check the answer key.

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42.

If

x

equals

\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}},

then:

x=1

0\lt x\lt1

1\lt x\lt2

x

is infinite

x\gt2

but finite

Answer: C

Concepts:radical recursion quadratic

Difficulty rating: 1530

Small hint:

The expression under the first radical contains another copy of

x

Big hint:

Solve

x=\sqrt{1+x}

and keep the nonnegative root

Solution:

The repeating tail gives

x=\sqrt{1+x},

so

x^2-x-1=0.

Since

x\ge0,

x=\frac{1+\sqrt5}{2},

which lies strictly between

1

and

2.

Thus, the correct answer is C.

Problem 42 in other years