Beyond infinity and more

Calculus Level 5

If sum of the infinite series $\dfrac12 + \dfrac28 + \dfrac{3}{16} + \dfrac{5}{32} + \dfrac{8}{64}+ \dfrac{13}{128} + \cdots = \dfrac{A}{B},$

where $$\gcd(A,B)=1$$, then find $$A+B$$.

Assumptions and Clarifications

• $$\dfrac{1}{4}$$ between $$\dfrac12$$ and $$\dfrac28$$ is deliberately omitted.
• With the exception of the missing $$\frac{1}{4}$$ term, the numbers in the numerator for the Fibonacci sequence and the numbers in the denominator are terms of a G.P.
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