An algebra problem by Akeel Howell

Algebra Level 2

x − 1 + x − 2 + x − 3 + ⋯ = x \large x^{-1}+x^{-2}+x^{-3}+\cdots = x If the positive root to the equation above is of the form a + b c \dfrac{a+\sqrt{b}}{c} , find a + b + c a+b+c .


The answer is 8.

Akeel Howell by Akeel Howell , 22, USA

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1 solution

Tom Engelsman
May 29, 2017

The above infinite geometric series on the LHS sums to 1 x 1 − 1 x = 1 x − 1 . \frac{\frac{1}{x}}{1 - \frac{1}{x}} = \frac{1}{x-1}. Hence, we arrive at the quadratic equation:

1 x − 1 = x ⇒ 0 = x 2 − x − 1 ⇒ x = 1 ± ( − 1 ) 2 − 4 ( 1 ) ( − 1 ) 2 = 1 ± 5 2 . \frac{1}{x-1} = x \Rightarrow 0 = x^2 - x - 1 \Rightarrow x = \frac{1 \pm \sqrt{(-1)^{2} - 4(1)(-1)}}{2} = \frac{1 \pm \sqrt{5}}{2}.

Since we are only interested in the positive root, we only allow x = 1 + 5 2 x = \frac{1 + \sqrt{5}}{2} , or the Golden Ratio. The desired sum is just 8.

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