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Harvard-MIT Math Tournament

Algebra Difficulty 4.7 AIME Prove it United States

Problem:
Find all values of $x$ that satisfy $x = 1 - x + x^{2} - x^{3} + x^{4} - x^{5} + \cdots$ (be careful; this is tricky).

Solution

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

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