Philip J. Fry inherited two-thirds of his genes from his mother and one-third from his grandmother. How is this possible?

In the following equations, F represents Philip, M his mom, D his dad, and G his grandmother.

1. F = 1/2*M + 1/2*D Fry gets half his genes from his mom and half from his dad.
2. D = 1/2*G + 1/2*F Dad gets half his genes from his mom (Grandma Mildred) and half from his dad (Fry himself).
3. 1/2*D = 1/4*G + 1/4*F Divide equation (2) in half.
4. F = 1/2*M + 1/4*G + 1/4*F Substitute equation (3) into equation (1).
Therefore, Fry gets half his genes from his mother, a quarter from Grandma Mildred, and a quarter from himself. This is where it gets tricky:
• F = 1/2*M + 1/4*G + 1/4*F Original equation.
• F = 1/2*M + 1/4*G + 1/4*(1/2*M + 1/4*G + 1/4*F) Substitute the original equation into itself.
• F = 1/2*M + 1/4*G + (1/8*M + 1/16*G + 1/16*F) Distribute the 1/4
• F = (1/2+1/8)*M + (1/4+1/16)*G + 1/16*F Group like terms.
• F = (1/2+1/8)*M + (1/4+1/16)*G + 1/16*F New equation.
• F = (1/2+1/8)*M + (1/4+1/16)*G + 1/16*{(1/2+1/8)*M + (1/4+1/16)*G + 1/16*F} Substitute the new equation into itself.
• F = (1/2+1/8)*M + (1/4+1/16)*G + {1/16*(1/2+1/8)*M + 1/16*(1/4+1/16)*G + 1/16*1/16*F} Distribute the 1/16.
• F = (1/2+1/8)*M + (1/4+1/16)*G + {(1/32+1/128)*M + (1/64+1/256)*G + 1/256*F} Distribute the 1/16 again.
• F = (1/2+1/8+1/32+1/128)*M + (1/4+1/16+1/64+1/256)*G + 1/256*F Group like terms.

Continue indefinitely to get:
F =
(1/2+1/8+1/32+1/128+1/512+1/2048...)*M +
(1/4+1/16+1/64+1/256+1/1024+1/4096...)*G +
(an infinitesimally small fraction)*F.

The sum (1/2+1/8+1/32+1/128+1/512+1/2048...) converges to 2/3. The sum (1/4+1/16+1/64+1/256+1/1024+1/4096...) converges to 1/3. Therefore, Fry's genetic makeup is 2/3 from his mother, 1/3 from his grandmother, and an infinitesimally small Y chromosome.

QED.