Maths Olympiad Prep

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Algebra Difficulty 4.7 AIME Prove it United States

Problem:
Find all values of xx that satisfy x=1−x+x2−x3+x4−x5+⋯x = 1 - x + x^{2} - x^{3} + x^{4} - x^{5} + \cdots (be careful; this is tricky).

Solution

Solution:
Multiplying both sides by 1+x1 + x gives (1+x)x=1(1 + x)x = 1, or x=−1±52x = \frac{-1 \pm \sqrt{5}}{2}. However, the series only converges for ∣x∣<1|x| < 1, so only the answer x=−1+52x = \frac{-1 + \sqrt{5}}{2} makes sense.

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