Quintic Equation?

Algebra Level 3

S = ∑ i = 0 ∞ ( − 1 ) i x i S = \sum_{i=0}^\infty (-1)^i x^i

If x x satisfies the equation x 5 = 1 − x x^5=1-x , which of the answer options is an expression for S S ?

x 2 − 1 x 2 + 1 \frac {x^2-1}{x^2+1} 1 + x 2 x \frac {1+x^2}{x} x − 1 x 2 + 1 \frac {x-1}{x^2+1} 1 − x 1 − x 2 \frac {1-x}{1-x^2}

Chris Sapiano by Chris Sapiano , 19, United Kingdom

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1 solution

Chew-Seong Cheong
Jun 28, 2019

S = ∑ i = 0 ∞ ( − 1 ) i x i = 1 − x + x 2 − x 3 + x 4 − x 5 + x 6 − x 7 + ⋯ = ( 1 − x ) + x 2 ( 1 − x ) + x 4 ( 1 − x ) + x 6 ( 1 − x ) + ⋯ = ( 1 − x ) ( 1 + x 2 + x 4 + x 6 + ⋯ ) Since x 5 = 1 − x ⟹ ∣ x ∣ < 1 , = 1 − x 1 − x 2 = 1 1 + x Maclaurin series applies. \begin{aligned} S & = \sum_{i=0}^\infty (-1)^i x^i \\ & = 1 - x + x^2 - x^3 + x^4 - x^5 + x^6 - x^7 + \cdots \\ & = (1-x) + x^2(1-x) + x^4(1-x) + x^6(1-x) + \cdots \\ & = (1-x)(1+x^2+x^4+x^6 + \cdots) & \small \color{#3D99F6} \text{Since }x^5=1-x \implies |x| < 1, \\ & = \boxed{\dfrac {1-x}{1-x^2}} = \frac 1{1+x} & \small \color{#3D99F6} \text{Maclaurin series applies.} \end{aligned}

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