The first three terms of a geometric progression are √2, √[3]2, √[6]2. Find the fourth term.

1961 AMC 12 Problem 12

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12.

The first three terms of a geometric progression are

\sqrt2,

\sqrt[3]2,

\sqrt[6]2.

Find the fourth term.

1

\sqrt[7]2

\sqrt[8]2

\sqrt[9]2

\sqrt[10]2

Write each radical as a power of

2

Check the differences between the successive exponents

The exponents are

\frac{1}{2},

\frac{1}{3},

and

\frac{1}{6}.

Each term is obtained by multiplying by

2^{-\frac{1}{6}},

so the next exponent is

\frac{1}{6}-\frac{1}{6}=0.

The fourth term is

2^0=1.

Thus, the correct answer is A.