Problem 13 · 2011 Math Kangaroo
Stretch
Algebra & Patterns
number-systems
How big is n if \(9^{n} + 9^{n} + 9^{n} = 3^{2011}\) holds true?
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Answer: A — 1005
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Hint 1 of 2
Three equal copies of 9ⁿ add to 3 times 9ⁿ.
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Hint 2 of 2
Rewrite everything as a single power of 3 and match exponents.
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Approach: rewrite as one power of 3
- 9ⁿ + 9ⁿ + 9ⁿ = 3 · 9ⁿ = 3 · (3²)ⁿ = 3 · 3^(2n) = 3^(2n+1).
- Set 3^(2n+1) = 3^2011, so 2n + 1 = 2011.
- Then 2n = 2010, giving n = 1005.
- So n = 1005, choice (A).
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