A different type of algebra problem

Algebra Level 5

∑ n = 1 9 x ( − 1 ) n = ∑ n = 1 13 x n \large \displaystyle\sum _{ n=1 }^{ 9 }{ { x }^{ (-1)^{ n } } } =\displaystyle\sum _{ n=1 }^{ 13 }{ { x }^{ n } }

How many real solutions does the above equation have?

Bonus : Find them (I don't think they have a nice closed form at all).


The answer is 2.

Hamza A by Hamza A , 19, USA

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1 solution

Yashas Ravi
May 13, 2020

Plug in small values (-2,-1,0,1,2) to see that the function intersects the x x -axis twice and then increases as x x approaches negative or positive infinity, so there are 2 roots.

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