Problem 11 · 2010 Math Kangaroo
Stretch
Algebra & Patterns
arithmetic-sequence
The numbers \(\sqrt{7}\), \(\sqrt[3]{7}\) and \(\sqrt[6]{7}\) are, in this order, consecutive terms of a geometric sequence. Determine the next term.
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Answer: E — 1
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Hint 1 of 2
Write each root as a power of 7 with a fractional exponent.
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Hint 2 of 2
In a geometric sequence the exponents form an arithmetic sequence.
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Approach: convert roots to fractional exponents
- The terms are 7^(1/2), 7^(1/3), 7^(1/6).
- The exponents 1/2, 1/3, 1/6 drop by 1/6 each step (common ratio 7^(−1/6)).
- The next exponent is 1/6 − 1/6 = 0, so the next term is 7^0 = 1.
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