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Hari is obsessed with cubics. He comes up with a cubic with leading coefficient 1, rational coefficients and real roots

0 < a < b < c < 1

. He knows the following three facts:

P(0) = -\frac{1}{8}

, the roots form a geometric progression in the order

a,b,c

, and

\sum_{k=1}^{\infty} (a^k + b^k + c^k) = \dfrac{9}{2}

The value

a + b + c

can be expressed as

\frac{m}{n}

, where

m,n

are relatively prime positive integers. Find

m + n

.
Proposed by Akshar Yeccherla (TopNotchMath)

Problem Statement