Problem 13 · 2016 Math Kangaroo
Hard
Algebra & Patterns
substitution
Which value does \(x_4\) take if \(x_1 = 2\) and \(x_{n+1} = x_n^{\,x_n}\) for \(n \ge 1\)?
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Answer: C — \(2^{2^{11}}\)
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Hint 1 of 2
Compute the terms one at a time, keeping everything as a power of 2.
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Hint 2 of 2
Track only the exponent: each step the new exponent is the old value times the old exponent.
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Approach: iterate, keeping powers of 2
- \(x_1 = 2\) and \(x_2 = 2^2 = 4\).
- \(x_3 = 4^4 = (2^2)^4 = 2^8\).
- \(x_4 = (2^8)^{2^8} = 2^{8 \cdot 256} = 2^{2048} = 2^{2^{11}}\), which is option C.
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